Switchings, realizations, and interpolation theorems for graph parameters

THEOREMS FOR GRAPH PARAMETERS

Received 17 March 2005 and in revised form 24 May 2005

Interpolation theorems on several graph parameters obtained in the past few years will be reviewed in this paper. Some simplified proofs are provided. Open problems in this direction are reviewed.

1. Introduction

Only finite simple graphs are considered in this paper. For the most part, our notation and terminology follows that of Bondy and Murty [4]. LetG=(V,E) denote a graph with vertex setV=V(G) and edge setE=E(G). We will use the following notation and terminology for a typical graphG. LetV(G)= {v1,v2,...,vn}andE(G)= {e1,e2,...,em}.

We use|S|to denote the cardinality of a setSand therefore we definen= |V|theorder ofGandm= |E|thesizeofG. To simplify writing, we writee=uvfor the edgeethat joins the vertexu to the vertexv. Thedegree of a vertexv of a graphG is defined as dG(v)= |{e∈E:e=uvfor someu∈V}|. The maximum degree of a graphGis usually denoted by∆(G). IfS⊆V(G), the graphG[S] is the subgraph induced bySinG. For a graphGandX⊆E(G), we denote byG−Xthe graph obtained fromGby removing all edges inX. IfX= {e}, we writeG−eforG− {e}. For a graphGandX⊆V(G), the graph G−Xis the graph obtained fromGby removing all vertices inXand all edges incident with vertices inX. For a graphGandX⊆E(G), we denote byG+Xthe graph obtained fromGby adding all edges inX. IfX= {e}, we simply writeG+eforG+{e}. Two graphs GandHare disjoint ifV(G)∩V(H)= ∅. For any two disjoint graphsGandH, we define G∪H(their union) byV(G∪H)=V(G)∪V(H) andE(G∪H)=E(G)∪E(H). We can extend this definition to a finite union of pairwise disjoint graphs, since the operation “∪” is associative. For a positive integer pand a graphG,pGis denoted for the union ofp copies ofG. A graphGis said to ber-regularif all of its vertices have degreer. A 3-regular graph is called acubic graph.

Let G be a graph of order n and let V(G)= {v1,v2,...,vn} be the vertex set of G. The sequence (dG(v1),dG(v2),...,dG(vn)) is called adegree sequence of G. A sequence d=(d1,d2,...,dn) of nonnegative integers is agraphic degree sequenceif it is a degree sequence of some graphG. In this case,Gis called arealizationofd.

Copyright©2005 Hindawi Publishing Corporation

An algorithm for determining whether or not a given sequence of nonnegative integers is graphic was independently obtained by Havel [22] and Hakimi [14]. We state their results in the following theorem.

Theorem1.1. Letd=(d1,d2,...,dn)be a nonincreasing sequence of nonnegative integers and denote the sequence

d2−1,d3−1,...,dd1+1−1,dd1+2,...,dn

=d. (1.1)

Thendis graphic if and only ifdis graphic.

LetGbe a graph and letab,cd∈E(G) be independent, whereac,bd /∈E(G). Put

Gσ(a,b;c,d)_{=G− {ab,cd}+{ac,bd}. (1.2)}

The operationσ(a,b;c,d) is called aswitching operation. It is easy to see that the graph obtained fromGby a switching has the same degree sequence asG. The following theo-rem has been shown by Havel [22] and Hakimi [14].

Theorem1.2. Letd=(d1,d2,...,dn)be a graphic degree sequence. IfG1andG2are any

two realizations ofd, thenG2can be obtained fromG1by a finite sequence of switchings.

As a consequence ofTheorem 1.2, Eggleton and Holton [10] defined in 1979the graph ᏾(d) of realizations ofdwhose vertices are the graphs with degree sequenced; two ver-tices being adjacent in the graph᏾(d) if one can be obtained from the other by a switch-ing. They obtained the following theorem.

Theorem1.3. The graph᏾(d)is connected.

The following theorem was shown by Taylor [38] in 1980.

Theorem1.4. For a graphic degree sequenced, letᏯ᏾(d)be the set of all connected real-izations ofd. Then the induced subgraphᏯ᏾(d)of᏾(d)is connected.

LetGbe the class of all simple graphs, a function f :G→Zis called agraph param-eter ifG∼=H, then f(G)= f(H). If f is a graph parameter andJ⊆G, f is called an interpolation graph parameter with respect toJif there exist integersxandysuch that

f(G) :G∈J=[x,y]= {k∈Z:x≤k≤y}. (1.3)

If f is an interpolation graph parameter with respect to J,{f(G) :G∈J}is uniquely determined by min(f,J)=min{f(G) :G∈J}and max(f,J)=max{f(G) :G∈J}. In the case whereJ=᏾(d), we simply write min(f,d) and max(f,d) for min(f,᏾(d)) and max(f,᏾(d)), respectively, and in the case whereJ=Ꮿ᏾(d), we write Min(f,d) and Max(f,d) for min(f,Ꮿ᏾(d)) and max(f,Ꮿ᏾(d)), respectively.

2. Interpolation theorems

f interpolate overJ? If f interpolates overJ, then{f(G) :G∈J}is uniquely determined by min(f,J)=min{f(G) :G∈J}and max(f,J)=max{f(G) :G∈J}; thus the second part of the interpolation theorems for graph parameters is to find the values of min(f,d) and max(f,d) for the corresponding interpolation graph parameters and this part is, in fact, the extremal problem in graph theory.

The interest in the interpolation properties of graph parameters was motivated by an open question posed by Chartrand during a conference held at Kalamazoo in 1980. He posed the following question: If a graphGcontains spanning trees havingmandn end-vertices, withm < n, doesGcontain a spanning tree withkend-vertices for every integerk withm < k < n? This question (which was answered affirmatively) led to a host of papers studying the interpolation properties of invariants of spanning trees of a given graph. Details can be found in [15,16,17,18,19,20,21]. Another interpolation theorem for the graph parameterχ+χwas obtained by Fink [12].

Letdbe a graphic degree sequence. It was shown that the graph᏾(d) is connected and its subgraph induced by the setᏯ᏾(d) is also connected.

Theorem2.1. For a graphGof degree sequencedand a switchingσif|f(G)−f(Gσ_{)| ≤1,} then f is an interpolation graph parameter with respect to᏾(d).

Theorem2.2. For a graphGof degree sequencedand a switchingσif|f(G)−f(Gσ_{)| ≤1,} then f is an interpolation graph parameter with respect toᏯ᏾(d).

Theorem2.3. LetJ⊆᏾(d)and the subgraph of᏾(d)induced byJbe connected. For a graphGof degree sequencedand a switchingσif|f(G)−f(Gσ_{)| ≤1, then f is an} interpo-lation graph parameter with respect toJ.

We will now review interpolation results on various graph parameters with respect to ᏾(d). Since our work started with the graph parameterχ, we first state the definition of χ.

Graph coloring takes its name from the map-coloring application. We assign labels to vertices when the numerical value of labels is unimportant, we call themcolorsto indicate that they may be elements of any set.

Ak-coloringof a graphG=(V,E) is a partition of its vertex setVasV1∪V2∪ ··· ∪

Vk such that no two vertices inVi(1≤i≤k) are adjacent. TheVi’s are called thecolor classes. A function f :V → {1, 2,...,k}such that f(v)=ifor eachv∈Vi(1≤i≤k) is called acolor function. IfGhas ak-coloring, it is said to bek-colorableand the minimum integerkfor whichGisk-colorable is called thechromatic numberofGand is denoted by χ(G). Ifχ(G)=k, we say thatGisk-chromatic.

Remark 2.4. In a proper coloring, each color class contains no edge, soGisk-colorable if and only ifGis ak-partite graph. Thus a graph is 2-colorable if and only if it is bipartite. Thus a graph containing an odd cycle must be at least 3-colorable.

We proved in [25] the following results.

ac,bd /∈E(G), ac,bd∈E(Gσ_{), andab,cd /∈E(G}σ_{). Letπ:V(G)→ {1, 2,...,χ(G)} be a} color function ofGsuch thatπ(a)=i,π(b)=j,π(c)=p, andπ(d)=q. Then we will de-fineπ:V(Gσ_{)→ {1, 2,...,χ(G),χ(G) + 1}byπ(c)=π(d)=χ(G) + 1 andπ(v)=π(v)} for allv∈V(Gσ_{)− {c,d}. Thus the graphG}σ_{is (χ(G) + 1)-colorable. This completes the}

proof.

A maximal complete subgraph of a graphGis called acliqueofG. The maximum order of clique ofGis called theclique numberofGand is denoted byω(G).

In general, there is no formula for the chromatic number of a graph. In fact, determin-ing the chromatic number of even a relatively small graph is often a challengdetermin-ing problem. However, lower bounds for the chromatic number of a graphGcan be given in terms of the clique number ofG. That is for any graphG,χ(G)≥ω(G).

We proved in [26]the following results on interpolation theorems onω.

Theorem2.6. LetGbe a graph and letσbe a switching onG. Then|ω(G)−ω(Gσ_{)| ≤1.} Proof. Letσ(a,b;c,d)=σ be a switching onG, such thatab,cd∈E(G),ac,bd /∈E(G). ThenGσ_{is the graph obtained fromGby deleting edgesab,cdand adding edgesac,bd.} Since verticesaandccannot lie in the same complete subgraph ofG,ω(Gσ_)≥ω(G)− 1. By symmetry of switching, we may assume thatω(G)≥ω(Gσ_{). Thusω(G}σ_{) is either} ω(G)−1 orω(G). In both cases, we have|ω(G)−ω(Gσ_{)| ≤1.} Acyclic graphis a graph containing no cycle as its subgraph. An acyclic graph is called a forest. Therefore, each component of an acyclic graph is a tree. Since a tree is connected, every two vertices in a tree are connected by a unique path.

Let G be a graph andF⊆V(G), F is called an induced forest of G, if the induced subgraphG[F] ofGcontains no cycle. For a graphG, we defineI(G) as

I(G) :=max|F|:Fis an induced forest inG. (2.1)

We proved in [28] the following results on interpolation theorems on the graph pa-rameterI.

Theorem2.7. IfS is any subset of vertices ofG such thatG[S]is a forest, andσ is any switching onG, thenGσ_{[S]contains at most one cycle.}

Corollary2.8. Ifσis any switching onG, then|I(G)−I(Gσ_{)| ≤1.}

The problem of determining the minimum number of vertices whose removal elimi-nates all cycles in a graphGis difficult even for some simply defined graphs. For a graph G, this minimum number is known as thedecycling numberofG, and is denoted byφ(G). It is easy to see that for a graphGof ordern,φ(G) +I(G)=n. Thus the interpolation result forφin᏾(d) is easily obtained.

It is clear thatα0(G) is a graph parameter andα0(G)=ω(G), for any graphG. Observe

that for a graphGand a switchingσonG,Gσ=Gσ_{. Thus the interpolation result forα0} in᏾(d) follows directly from the graph parameterω.

A vertex of a graphG=(V,E) is said to cover the edges incident with it. Avertex cover of a graphGis a set of vertices covering all the edges ofG. The minimum cardinality of a vertex cover of a graphGis called thevertex covering numberofGand is denoted by β0(G).

Within a set of people, some pairs are compatible as roommates; under what condition can we pair them all up? Many applications of graphs involve such pairings. Problem of job assignment with qualified applicants is also an example of matching.

A subsetMof the edge setEof a graphG=(V,E) is anindependent edge setor match-inginGif no two distinct edges inMhave a common vertex. A matchingMismaximum inG if there is no matchingMof Gwith|M|>|M|. The cardinality of a maximum matching ofGis denoted byα1(G) and is called thematching numberofG.

There is an analogous covering concept for edges.

An edge of a graphG=(V,E) is said to cover the two vertices incident with it. An edge coverof a graphGis a set of edges covering all the vertices ofG. The minimum cardinality of an edge cover ofGis called theedge covering numberofGand is denoted byβ1(G).

Theorem2.9 [31]. IfGis a graph withα1(G)=α1andσis a switching onG, thenα1(Gσ)≥

α1−1.

Proof. LetMbe an independent set of edges ofEwith|M| =α1(G). Letσ(a,b;c,d)=σ

be a switching onG. If{ab,cd} ∩M= ∅, then|M| = |Mσ_{|. If{ab,cd} ⊆M, then|M| =} |Mσ_{|. Finally, ifMcontains exactly one edge from the set{ab,cd}, then|M}σ_{| = |M| −1.}

Therefore,α1(Gσ)≥α1−1.

Corollary2.10 [31]. Ifσis a switching onG, then|α1(G)−α1(Gσ)| ≤1.

Letnbe a positive integer. Thestarof ordern+ 1 is the complete bipartite graphKn,1.

The edges covered by one vertex in a vertex cover are the edges incident to it; they form a star. The vertex cover problem can be described as covering the edge set with the fewest number of stars. This is equivalent to our next graph parameter.

Adominating setof a graphG=(V,E) is a subsetDofVsuch that each vertex ofV−D is adjacent to at least one vertex ofD. Thedomination numberγ(G) is the cardinality of a minimal dominating set with the least number of elements.

Theorem2.11 [31]. IfGis a graph withγ(G)=γandσis a switching onG, thenγ(Gσ_)≤ γ+ 1.

Proof. LetDbe a dominating set of vertices ofV with|D| =γ(G). Letσ(a,b;c,d)=σ be a switching onG. If{a,b,c,d} ∩D= ∅, then|D| = |Dσ_{|. If{a,b,c,d} ⊆D, then|D| =} |Dσ_{|. Ifa∈Dandb,c,d∈V−D, thenD∪ {b}is a dominating set ofG}σ_{. Ifa,b∈Dand} c,d∈V−D, thenDis a dominating set ofGσ_{. Ifa,c∈D, thenD∪ {b}is a dominating} set ofGσ_{. Finally, ifa,b,c∈Dandd∈V−D, thenD∪ {d}is a dominating set ofG}σ_.

Thusγ(Gσ_{)≤γ(G) + 1.}

Proof. Since a switching is symmetric, we may assume thatγ(G)≤γ(Gσ_{). ByTheorem} 2.11,γ(Gσ_{) is eitherγ(G) + 1 orγ(G). In both cases, we have|γ(G)−γ(G}σ_{)| ≤1.} Gallai [13] and Norman-Rabin [23] proved the following results concerning relation-ship betweenα0andβ0, andα1, andβ1, respectively.

Theorem2.13 [13]. For any graphGof ordern,α0+β0=n.

Theorem2.14 [23]. For any graphGof ordernandδ≥1,α1+β1=n.

The interpolation property of the graph parametersβ0 andβ1follows from the last

two theorems. Thus combining the results in this section, we can conclude the following results.

Theorem 2.15. Let d=(d1,d2,...,dn), d1≥d2≥ ··· ≥dn≥1 be a graphic degree se-quence. Thenχ,ω,I,φ,α0,α1,β0,β1, andγare interpolation graph parameters with respect

to᏾(d).

Theorem2.16. Let f ∈ {χ,ω,I,φ,α0,α1,β0,β1,γ}. Then for any graphic degree sequenced,

there exist integersa:=a(f)andb:=b(f)such thatdhas a realizationGwith f(G)=cif and only ifcis an integer satisfyinga≤c≤b.

3. Extremal graph parameters

In this section, we discuss the second part of interpolation theorems for graph parame-ters. We first recall what we mean by an extremal problem in graph theory. Anextremal problemasks for minimum and maximum values of a function over a class of objects. Remark 3.1. Proving thatA is the minimum of f(G) for graphs in a class J requires showing two things:

(1) f(G)≥Afor allG∈J; (2) f(G)=Afor someG∈J.

The proof of the bound must apply to everyG∈J. For equality it suffices to obtain an example inJwith the desired value of f.

Changing “≥” to “≤” yields the criteria for a maximum.

It will be seen that we have produced several results in the extremal graph theory for graph parametersχ,ω,I,φ, andα1in the class of allr-regular graphs of ordernand some

other related classes.

In their proof, they used the concept of thecovering numberof a graphG. Acovering of a graphGis a partitionP ofV(G) such that for eachVi∈P, the induced subgraph G[Vi] inGis a complete graph. The covering number ofG, denoted byc(G), is defined by

c(G) :=min|P| |Pis a covering of G. (3.1)

It is easy to see that for any graphG,c(G)=χ( ¯G). We extended their result by improving the upper bound in [24].

Let f be an interpolation graph parameter with respect to᏾(d). It is clear that{f(G) : G∈᏾(d)}is uniquely determined by min(f,d) and max(f,d). We will discuss in this section those extremal values for various kinds of graph parameters. For simplicity rea-son, we will consider the class of regular graphs and some other related classes of graphs.

For the graph parameterχ, we obtained in [25] the following results. Theorem3.3. Ifr≥2andn≥2r, then

minχ,rn₌

  

2 ifnis even,

3 ifnis odd. (3.2)

Theorem3.4. Ifr≥2, then

minχ,rr+1_=maxχ,rr+1_{=r+ 1,}

minχ,rr+2_=maxχ,rr+2₌r+ 2

2 .

(3.3)

The proofs of Theorems3.3and3.4were based on graph constructions.

Theorem3.5. For anyr≥4and odd integerssuch that3≤s≤r, letqandtbe integers satisfyingr+s=sq+t, 0≤t < s. Then

minχ,rr+s=

        

q ift=0, q+ 1 if1≤t≤s−2, q+ 2 ift=s−1.

(3.4)

Theorem3.6. For any even integerr≥6and any even numberssuch that4≤s≤r, letq andtbe integers satisfyingr+s=sq+t,0≤t < s. Then

minχ,rr+s₌

  

q ift=0,

q+ 1 ift≥2. (3.5)

Theorem3.7. Letr≥2. Then (1)

maxχ,r2r_{=r, (3.6)}

(2)

maxχ,r2r+1₌

  

3 ifr=2,

r ifr≥4, (3.7)

(3)

maxχ,rn=r+ 1 forn≥2r+ 2. (3.8)

Brooks’ theorem [5] and graph constructions were applied to prove this theorem. Theorem3.8. For anyrandssuch that3≤s≤r−1,

(1)

maxχ,rr+s≥r+s

2 ifr+sis even, (3.9)

(2)

maxχ,rr+s≥r+s−1

2 ifr+sis odd. (3.10)

Purely graph constructions were used in the proof of this theorem. The exact values of max(χ,rn_{) are not easy to obtain whenr+ 3≤n≤2r−1.Theorem 3.2was proved by} Erd˝os and Gallai [11] to provide an upper bound in the class of connected proper regular graphs of ordern.

It should be noted that the bound given byTheorem 3.2does not depend onrand this bound may be very far from the actual values of the chromatic numbers. Also the exact value of max(χ,rr+3_{) was obtained byTheorem 3.2. We gave in [24] the following}

definition.

Let j be a positive integer. AnF(j)-graphis a (j−1)-regular graphGof minimum order f(j) with the property thatχ(G)> f(j)/2.

It is easy to see thatF(3)-graph is a unique graphC5and f(3)=5. We will see later

thatF(j)-graphs, j≥5, are not unique.

We foundF(j)-graphs for all odd integersjas we state in the following theorem. Theorem3.9 [24]. For odd integer jwith j≥3, f(j)=(5/2)(j−1)if j≡3(mod 4)and

f(j)=1 + (5/2)(j−1)if j≡1(mod 4).

Theorem 3.10 [24]. Anyr-regular graph of order n withn−r=j odd and j≥3 has chromatic number at most((f(j) + 1)/2f(j))·n, and this bound is achieved precisely for those graphs with complement equal to a disjoint union ofF(j)-graphs.

Problem 3.11. Find anF(j)-graph for even integer j≥4.

Problem 3.12. Find the value of max(χ,rr+j_{) for even integerjand 4≤j≤r−2.} For the graph parameterω, we completely answered in [26] the second part of inter-polation theorem with respect to the class ofr-regular graphs of ordernas stated in the following results.

Theorem3.13. LetGbe a graph of ordernand maximum degree∆. Thenω(G)≥n/(n− ∆).

Since the complete graph Knis a clique, min(ω,rr+1)=max(ω,rr+1)=r+ 1. Given positive integersnandk withk≤n, there exists a connected graphG of ordernwith ω(G)=k. However, ifGis regular, we have the following remarkable result.

Theorem 3.14. Let d=rn _{be a graphic degree sequence with r+ 2≤n≤2r+ 1. Then} max(ω,rn_{)= n/2.}

Theorem3.15. For anyr≥6and odd integerssuch that5≤s < r, letqandtbe integers satisfyingr+s=sq+t,0≤t < s. Then

minω,rr+s₌

        

q ift=0, q+ 1 if1≤t≤s−2, q+ 2 ift=s−1.

(3.11)

Theorem3.16. For any even integerr≥6and any even numberssuch that4≤s≤r, letq andtbe integers satisfyingr+s=sq+t,0≤t < s. Then

minω,rr+s₌

 

q_{q+ 1} if_ift_t=_≥0,_2. (3.12)

Observe that results of Theorems3.15and3.16, respectively, coincide with Theorems 3.5and3.6above.

For graph parameter I, we found in [27] a lower bound of min(I,d) by usingthe probabilistic method. In particular, we proved that ifGis a graph with degree sequence d=(d1,d2,...,dn),d1≥d2≥ ···dn≥1, then

I(G)≥2 n

i=1

1

di+ 1. (3.13)

As an application, we found all minimum values of the order of maximum induced forest in a (∆,n)-graph.

First of all we would like to introduce basic tools from discrete probability theory which will be used for calculation of some bounds of graph parameters.

Let (Ω,p) be a finiteprobability space, whereΩis a finite set and pis a probability function that maps fromΩinto the interval [0, 1] with

ω∈Ωp

Arandom variableXonΩis a mappingX:Ω→R. We define a probability space on the image setX(Ω) as

p(X=x) := X(ω)=x

p(ω). (3.15)

TheexpectationE(X) ofXis

E(X)= ω∈Ω

p(ω)X(ω). (3.16)

Now supposeX andY are two random variables onΩ, then the sumX+Y is again a random variable, sharing a propertyE(X+Y)=E(X) +E(Y).

Clearly, this can be extended to any finite linear combination of random variables (the linearity of expectation). Note that it is not necessary that the random variables are “independent.”

Thus we have the following theorems.

Theorem3.17 [27]. LetGbe a graph with degree sequence

d=d1,d2,...,dn

, d1≥d2≥ ···dn≥1. (3.17)

Then

I(G)≥2 n

i=1

1

di+ 1. (3.18)

Proof. LetGbe an arbitrary graph on the vertex setV= {v1,v2,...,vn}. Denote the degree ofvibydi. We choose with equal probability 1/n! a random permutationπ=π1π2···πn ofV and construct the following setFπ. We putπiintoFπ if and only ifπi is adjacent to at most oneπj (j < i). It is clear thatFπ is an induced forest ofG. LetX(π)= |Fπ| be the corresponding random variable. We haveX=n

i=1Xi, whereXiis the indicator random variable of the vertexvi, in other wordsXi=1 ifvi∈FπandXi=0 ifvi∈/ Fπ. In order to calculateE(Xi), we first calculate the number of permutationsπ which contain the vertexvi. Sincevihas di neighbors, we partitionn! permutations on V into

n di+1

classes according to di+ 1 positions that vi and its neighbors appeared. It is clear that each class contains (di+ 1)!(n−di−1)! permutations and 2di!(n−di−1)! of which have the property thatvi∈Fπ. Thus

EXi

= π

p(π)Xi(π)=_d2

i+ 1. (3.19)

Therefore,

E(X)= n

i=1

EXi

=2 n

i=1

1

di+ 1. (3.20)

Corollary3.18. If the average degree ofGis at mostd, thenI(G)≥2n/(d+ 1). We now have the following corollary as a result of Caro [7].

Corollary3.19. LetGbe a graph with degree sequence

d=d1,d2,...,dn

, d1≥d2≥ ···dn≥0. (3.21)

Then

α0(G)≥

n

i=1

1

di+ 1, (3.22)

whereα0(G)is the independence number ofG.

Proof. The proof follows easily from the fact that a forest is bipartite. Corollary3.20. LetGbe a graph with degree sequence

d=d1,d2,...,dn

, d1≥d2≥ ···dn≥0. (3.23)

Then

ω(G)≥ n i=1

1

n−di, (3.24)

whereω(G)is the clique number ofG.

Proof. The proof follows from the fact thatω(G)=α0(G).

LetG be a graph. The problem of determining the decycling numberφ(G) of G is equivalent to finding the greatest order of an induced forestI(G) of the graphG and the sum of the two numbers equals the order of the graph. Observe that for a minimum decycling setSof a graphG, ifv∈S, then there exists a connected componentCofG−S such thatvis adjacent to at least two vertices ofC. Thus∆(G[S])≤∆(G)−2. With this observation, we find that ifGis anr-regular graph andSis a minimum decycling set of G, the graphG[S] may not be an (r−2)-regular graph. This causes a difficulty in finding max(φ,rn_{) if we consider only the class of regular graphs. It is reasonable to enlarge the} class of regular graphs into the following class of graphs. Let∆be a nonnegative integer and letnbe a positive integer such thatn≥∆+ 1. LetG(∆,n) be the class of all graphs of ordernand of maximum degree∆.The (∆,n)-graphis a graph havingG(∆,n) as its vertex set and two such graphs being adjacent if one can be obtained from the other by either adding or deleting an edge.

Lemma3.21 [27]. The(∆,n)-graph is connected.

Proof. For any graphG∈G(∆,n), ifF=K1,∆∪(n−∆−1)K1andG=F,Gcan be

Theorem3.22 [27]. Letd=(d1,d2,...,dn),d1≥d2≥ ··· ≥dn≥1be a graphic degree sequence andd1+ 1≤n≤2d1+ 1. Then

(1) min(I,d)=2if and only ifd1=d2=d3= ··· =dnandn=d1+ 1,

(2)ifddoes not have a complete graph as its realization, thenmin(I,d)=3if and only ifdhas a union of stars as its realization.

Theorem3.23 [27]. Letn=(∆+ 1)q+t,0≤t≤∆. Then (1) min(I,G(∆,n))=2qift=0,

(2) min(I,G(∆,n))=2q+ 1ift=1, (3) min(I,G(∆,n))=2q+ 2if2≤t≤∆.

We obtained in [28] the values of max(I,rn_{), for allrandnas stated in the following} theorems.

Theorem3.24.

max(I,rn)=

    

n−r+ 1 ifr+ 1≤n≤2r−1,

nr−2 2(r−1)

ifn≥2r. (3.25)

The values of min(I,rn_{), for allrandn, were obtained in [33] in terms of the graph} parameterφas stated in the following theorems. Note that min(I,rn_{) + max(φ,r}n_)=n. Theorem3.25. Forr≥3, andn=r+j,1≤j≤r+ 1,

(1) min(I,rn_{)=2if and only ifn=r+ 1,} (2) min(I,rn_{)=3if and only ifn=r+ 2,}

(3) min(I,rn_{)=4for all even integersn,r+ 3≤n,}

(4) min(I,rn_{)=4for all odd integersn,r+ 3≤nandn≥f(j),} (5) min(I,rn_{)=5for all odd integersn,r+ 3≤nandn < f(j),} where

f(j)=5

2(j−1) ifj≡3(mod 4), f(j)=1 +5

2(j−1) if j≡1(mod 4).

(3.26)

Theorem3.26. Forn≥2r+ 2andr≥3, writen=(r+ 1)q+t,q≥2and0≤t≤r. Then (1) min(I,rn_)=2qift=0,

(2) min(I,rn_{)=2q+ 1ift=1,} (3) min(I,rn_{)=2q+ 2if2≤t≤r−1,} (4) min(I,rn_{)=2q+ 3ift=r.}

In [32], we investigated the values of min(α1,rn) and max(α1,rn) for allrandn. It is

easy to see that min(α1, 0n)=max(α1, 0n)=0 and min(α1, 12n)=max(α1, 12n)=n.

Be-cause of this fact, from now on we will considerr≥2 andn≥r+ 1. We first investigate the value of max(α1,rn) by the following theorem.

Proof. Since there is a well-known result by Dirac (cited from [4]) that anr-regular graph of ordernwithr≥n/2 is Hamiltonian, we need to consider only whenr < n/2. It is easy to construct a Hamiltonian graph with 10 or fewer vertices. It is also easy to construct such a graph withr=2 or 3. LetX= {x0,x1,...,xt−1}andY= {y0,y1,...,yt−1}, where

t≥5. For an integerrwith 1≤r≤t, take the edge set

E=xiyi+j:i=0, 1, 2,...,t−1, j=0, 1, 2,...,r−1

, (3.27)

where all subscripts are taken modt. It is clear that the graphB(r;X,Y)=(X∪Y,E) is anr-regular bipartite graph and it is Hamiltonian ifr≥2. Suppose thatn=2t+ 1 andr is an even integer with 4≤r≤t, the graphG=(X∪Y∪ {u},E), whereE=[E(B(r− 2;X,Y))− {xiyi:i=0, 1, 2,...,r/2}]∪ {xixi+1,yiyi+1:i=0, 1, 2,...,t−1} ∪ {uxi,uyi:i=

0, 1, 2,...,r/2}, is anr-regular Hamiltonian graph.

We will use the generalized result of Tutte and the result of Wallis [41] to obtain all values of min(α1,rn).

A 1-factorof a graphGis a 1-regular spanning subgraph ofG. A 1-factorizationofGis a set of pairwise edge-disjoint 1-factors which together contain each edge ofG.

It is well known thatK2n andKn,n have 1-factorizations for all positive integersn. The question of which graphs contain 1-factors is one that has attracted considerable attention. For a comprehensive review, we refer to the survey of Akiyama and Kano [1] and of Wallis in [9].

A necessary and sufficient condition for a graph to have a perfect matching was ob-tained by Tutte [40]. A component of a graph isoddorevenaccording as it has an odd or even number of vertices. We denote byO(G) the number of odd components ofG. The following theorem is due to Tutte [40]

Theorem3.28. A graphGhas a perfect matching if and only if

O(G−S)≤ |S| ∀S⊆V. (3.28)

Berge [3] generalized Tutte’s result and it makes easier for application.

Theorem 3.29 [3]. The number of edges in a maximum matching of a graph G is (1/2)(|V(G)| −d), whered=maxS⊆V(G){O(G−S)− |S|}.

LetF(r,d) be the minimum order of anr-regular graphGwithα1(G)=(1/2)(|V(G)| −

d). It is clear that|V(G)| ≡d(mod 2). Wallis [41] foundF(r, 2) for all r≥3. In other words, he proved the following theorem.

Theorem3.30. LetGbe anr-regular graph with no1-factor and no odd component. Then

_V(G)≥         

3r+ 7 ifris odd,r≥3, 3r+ 4 ifris even,r≥6, 22 ifr=4.

(3.29)

Suppose thatGis anr-regular graph withα1(G)=(1/2)(|V(G)| −d). There exists a

k-subsetKofV(G) such thatO(G−K)=k+d. Ifk=0, thenris even,Gcontainsdodd components, and each component ofGhas order at leastr+ 1. Supposek≥1 andG−K has an odd component withp vertices, where pis less than or equal tor. The number of edges within the component is at most (1/2)p(p−1). This means that the sum of degree of these pvertices inG−K is at most p(p−1). ButGis anr-regular graph, so the sum of these p vertices inGis pr. The number of edges joining to the component Kmust be at leastpr−p(p−1). For a fixedrand for integer psatisfying 1≤p≤r, the function f(p)=pr−p(p−1), 1≤p≤r, has minimum value f(1)= f(r)=r. So any odd component withror less vertices is joined toKbyror more edges.

Suppose that there areO+odd components ofG−K with more thanrvertices and

O−odd components with less than orrvertices. Thus

O++O−=k+d, O++rO−≤kr. (3.30)

From these two relations, we haveO+≥ rd/(r−1) =d+d/(r−1)andk≥ d/(r−

1). Thus we have the following results.

Theorem3.31 [32]. Letrbe an even integer,r≥2. ThenF(r,d)=d(r+ 1).

Corollary3.32 [32]. Letrbe an even integer,r≥2. Ifn=(r+ 1)d+e,0≤e≤r, then min(α1,rn)=dr/2 +(1 +e)/2.

Suppose thatr is odd and r≥3. LetGbe an r-regular graph of order nsuch that α1(G)=(1/2)(n−d). Thendmust be even. Putd=2q. There exists a nonempty subset

KofV(G) of cardinalityksuch thatO(G−K)=k+ 2q. By (3.30), we have

n≥k+ (r+ 2)O+≥

₂

q r−1

+ (r+ 2)

2q+

₂

q r−1

=

₂

q r−1

(r+ 3) + 2q(r+ 2). (3.31)

Wallis [41] definedG(x,y) to be a graph withx+yvertices,xbeing of degreex+y−3 and y of degreex+y−2. G(x,y) exists if and only if yis even and y≥2. It is noted that for any graphG(x,y), it has yvertices of degreerandxvertices of degreer−1. Let xi,yi,i=1, 2,...,m, be integers such thatG(xi,yi) exists for alli=1, 2,...,m. We construct the graph

Gx1,y1∗Gx2,y2∗ ··· ∗Gxm,ym (3.32)

from disjoint copies of the graphs by inserting a new vertex, sayu, and joininguto all vertices ofG(xi,yi) which have the smallest degree, fori=1, 2,...,m.

Using this notation, we see that for an odd integerr≥3 andq=1, 2,..., (r−1)/2, for any odd positive integersai,i=1, 2,..., 1 + 2q, whose sum isr,

Gq=G

a1,r+ 2−a1

∗Ga2,r+ 2−a2

∗ ··· ∗Ga1+2q,r+ 2−a1+2q

is anr-regular graph on (r+ 2)(1 + 2q) + 1 vertices withα1(Gq)=(1/2)(|V(Gq)| −2q). We have the following results.

Theorem3.33 [32]. For an odd integerr≥3, then

(1)F(r, 2q)=(r+ 2)(1 + 2q) + 1, forq=1, 2,..., (r−1)/2,

(2)ifq=((r−1)/2)s+t,0≤t <(r−1)/2, thenF(r, 2q)=sF(r,r−1) +F(r, 2t), where F(r, 0)=0.

Corollary3.34 [32]. Letrbe an odd integer,r≥3. IfF(r, 2q)≤n < F(r, 2(q+ 1)), then min(α1,rn)=(1/2)(n−2q).

4. Induced subgraphs of᏾(d)

It was observed by Eggleton and Holton [10] that the graph ᏾(d) is connected. It is natural to investigate connected subgraphs of᏾(d).

LetP be a property which a graph may possess. Denote by᏾(d;P) the subgraph of the graph᏾(d) induced by those vertices which correspond to graphs with propertyP. PropertyPfor which᏾(d;P) is connected is calledcomplete. If a propertyPis complete, we may find all the graphs of a given degree sequence with the property by switching constrained to graphs with the property.

Colbourn [8] showed that the property of being a tree is complete and Syslo [37] extended this to the property of being unicyclic. Taylor [38] generalized these results by showing that the property of being connected is complete. The property of being 2-connected was shown to be complete also by Taylor [39].

By using results inSection 2and the results of Taylor, we immediately obtain the fol-lowing interpolation theorems.

Theorem4.1. Let f ∈ {χ,I,φ,ω,α0,α1,β0,β1,γ}. Then for any graphic degree sequenced,

there exist integersa:=a(f)andb:=b(f)such thatdhas a connected realizationGwith f(G)=cif and only ifcis an integer satisfyinga≤c≤b.

Theorem4.2. Let f ∈ {χ,I,φ,ω,α0,α1,β0,β1,γ}. Then for any graphic degree sequenced,

there exist integersa:=a(f)andb:=b(f)such thatd has a2-connected realizationG with f(G)=cif and only ifcis an integer satisfyinga≤c≤b.

Let f be an interpolation graph parameter with respect to᏾(d). Then{f(G) :G∈ ᏾(d)}is uniquely determined by min(f,d) and max(f,d). In particular, if f =χ, we can define thechromatic range,χ(d), as the interval of integers as

χ(d) :=[a,b]= {c∈Z:a≤c≤b}, (4.1)

wherea=min(χ,d) andb=max(χ,d). The following results concerning connected sub-graphs of᏾(d) were obtained in [35].

We consider the problem of determining the structure of induced subgraphs of the graph of realizations of a degree sequencedwith prescribed chromatic number. We ob-tained some significant results whendis a regular degree sequence.

Theorem4.3. Any graphGsatisfiesχ(G)≤1 +∆(G), with equality holds if and only if either of the following holds:

(1)some component ofGis the complete graphK∆+1, where∆=∆(G);

(2)some component ofGis an odd cycle, and∆(G)=2.

Note that Brooks’ theorem impliesb=max(χ,d)≤1+∆, where∆=max{d1,d2,...,dn}.

Thus for a regular degree sequencern_{, we haveb=max(χ,r}n_{)≤1 +r. We found in [25]} the corresponding value ofbfor all values ofrandnexcept the cases whenrandnare even andr+ 4≤n≤2r−2. In general, the Brooks bound may be very far from the actual value.

For eachc∈χ(d), let᏾(d; χ=c) denote the subgraph of᏾(d) induced by the ver-tices corresponding to graphs with chromatic numberc. Similarly for anyc∈χ(d), let ᏾(d;χ≤c) denote the subgraph of᏾(d) induced by the vertices corresponding to graphs with chromatic number p≤c. We consider the problem of determining the structure of induced subgraph᏾(d; χ=c) and᏾(d; χ≤c). In general, what is the structure of ᏾(d;χ=c) and of᏾(d;χ≤c)? In particular, are these graphs connected? If᏾(d;χ=c) is connected, it must be possible to generate all realizations ofdwith chromatic number cby beginning with one such realization and applying a suitable sequence of switchings producing only graphs with chromatic numberc. The same applies if᏾(d;χ≤c) is con-nected.

Note thatχ(0n_{)= {1},χ(1}n_{)= {2}(nis even),χ(2}4_{)= {2},χ(2}n_{)= {2, 3}ifnis even} andn≥6, andχ(2n_{)= {3}ifnis odd andn≥3. Forχ(3}n_{), we haveχ(3}4_{)= {4},χ(3}6₎₌

{2, 3}, andχ(3n_{)= {2, 3, 4}ifnis even andn≥8.}

LetG be a graph with χ(G)=k and letσ be a switching onG.σ is called a k-safe switchingifχ(Gσ_{)=k. A sequenceσ1,σ2,...,σ}

tof switchings is called a sequence ofk-safe switchings if for eachi,i=1, 2,...,t,χ(Gσ1σ2···σi₎=_k.

Theorem4.4 [35]. Ifr≥3andmax(χ,rn_{)=r+ 1, then the graphs᏾(r}n_{;χ=r+ 1)and} ᏾(rn_{;p≤r)are connected.}

Proof. Note that ifr≥3, then max(χ,rn_{)=r+ 1 if and only ifn=r+ 1 orn≥2r+ 2.} Moreover, ifGis a realization ofrn_{andχ(G)=r+ 1 if and only ifGhasK}

r+1as a

compo-nent, the theorem is true forn=r+ 1. Forn≥2r+ 2, letG1andG2be any two realizations

ofrn_{such thatχ(G1)=χ(G2)=r+ 1. ThusG1=K}

r+1∪H1andG2=Kr+1∪H2. SinceH1

andH2arer-regular graphs of ordern−r−1,H2 can be obtained fromH1 by a finite

number of switchings. ThusG2can as well be obtained by those switchings. Furthermore,

it is easy to observe that those switchings are (r+ 1)-safe switchings.

LetGbe a realization ofrn_{such thatχ(G)≤r. IfGis disconnected, thenGdoes not} containKr+1as its component. Thus there exists a suitable sequence of switchings which

transformsG to a connected realization ofrn _{such that each resulting graph obtained} in this transformation will have chromatic number less than or equal tor. By using the

result by Taylor [38], the proof is complete.

LetCmbe the cycle of orderm. ThusCmexists for all integerm≥3, we callCmodd cycle oreven cycleaccording to whethermis odd or even. A realization of 2n_{can be written} ast_i=1Cni, where

_t

only if for alli,niis even. It is clear that for any two even cyclesCrandCs, there is a 2-safe switching which transforms these two cycles toCr+s. Thus᏾(2n;χ=2) is connected. The corresponding result can be obtained for the graph᏾(2n_{; 3) with only one exception.} Theorem4.5 [35]. If3∈χ(2n_{), then the graph᏾(2}n_{; χ=3)is connected if and only if} n=10.

Proof. Observe that ifnis odd andn≥3, thenχ(2n_{)= {3}. Thus the graph᏾(2}n_;χ= 3) is connected. Ifn=10, we cannot transform 2C5 to 2C3∪C4 without passingC10.

Thus the graph᏾(21_{0; χ=3) is not connected. Forn≥12, let}t

i=1Cni be a realization

of 2n_{and suppose that n1 is odd. Then there is a sequence of 3-safe switchings which} transformst_i=1Cni toCn1∪Cn−n1. Sincen≥12,C5∪Cn−5can be transformed toC5∪ C3∪Cn−8and toC3∪Cn−3. Ifmis odd andm≥7, thenCm∪Cn−mcan be transformed toC3∪Cm−3∪Cn−mand then toC3∪Cn−3. The proof is complete.

Theorem 4.6 [42, page 53]. Let G andH be bipartite graphs with bipartition (X,Y). IfdG(v)=dH(v)for allv∈X∪Y, then there is a sequence of2-safe switchings that trans-formsGintoH.

The following theorem is a consequence ofTheorem 4.6.

Theorem4.7 [35]. If2∈χ(rn_{), then the graph᏾(r}n_{;χ=2)is connected.} Theorem4.8 [35]. Ifc∈χ(3n_{), then the graph᏾(3}n_{;χ=c)is connected.}

Proof. We have already proved whenc=2, 4. Note thatG∈᏾(3n_{) withχ(G)=3 if and} only ifGdoes not containK4 as its component andGhas an odd cycle as its subgraph.

It is easy to check that the theorem is true ifn≤8. LetG be a cubic graph of order n≥10 withχ(G)=3. We first suppose thatGhas a triangleTwithV(T)= {x,y,z}and E(T)= {xy,yz,xz}. Ifxandyhave a common neighborv∈G−T, then there exists an edgeabinG−Tindependent of the edgexv. Thus there is a 3-safe switchingσsuch that Gσ_{contains a triangleTwhere any two vertices ofThave no common neighbor inG−T.} Now suppose thatGcontains an odd cycleCof smallest orderk≥5. Letx,ybe two adjacent vertices in the cycle. Thusxandyhave no common neighbor inG. Letzbe the neighbor of yinG−C. Then there exists a 3-safe switchingσ such thatGσ _{contains a} triangleT= {x,y,z}. Thus for a graphG∈᏾(3n_{) andχ(G)=3, there is a sequence of} 3-safe switchings which transformsGtoGsuch thatGcontains a triangleT= {x,y,z} andG−Tis a graph having exactly 3 vertices of degree 2. SinceGis an arbitrary graph

in᏾(3n_{), the proof is complete.}

Conjecture4.9. Ifc∈χ(rn_{), then the graph᏾(r}n_{;χ=c)is connected.}

For the graph parameterφ, we investigated the values ofφ(G) whereGruns over the class of cubic graphs in [34]. As an application, we were able to answer a problem asked by Bau and Beineke [2].

of induced subgraph᏾(d; φ=c). In general, what is the structure of᏾(d; φ=c)? In particular, are these graphs connected? If᏾(d;φ=c) is connected, it must be possible to generate all realizations ofdwith decycling numbercby beginning with one such realiza-tion and applying a suitable sequence of switchings to produce only graphs with decycling numberc.

Bau and Beineke posed the following problem: Which cubic graphsGwith|G|=2n satisfyφ(G)= (n+ 1)/2?

We answered the problem by finding all cubic graphs G of order 2n with φ(G)= (n+ 1)/2. Furthermore, we proved that the induced subgraph᏾(32n_{;φ= (n+ 1)/2)} is connected. We proved in the same paper that if᏾(32n_{) is the class of all cubic graphs} of order 2n, then min{φ(G) :G∈᏾(32n_{)} = (n+ 1)/2. Thus to answer the problem is} equivalent to finding all cubic graphs of order 2nin᏾(32n_{) having minimum cardinality} of decycling set.

Conjecture4.10. Ifc∈φ(rn_{), then the graph᏾(r}n_{;φ=c)is connected.}

5. New approach on graph parameters

We have introduced a new area of research on graph parameters. Let f be a graph pa-rameter and let f(n,k,r) be the set of all connectedr-regular graphsGof ordernand f(G)=k. We consider the problem of determining the set of all integers r for which f(n,k,r)= ∅.

Letχ(n,k,r) be the set of all connected,r-regular,k-chromatic graphs onnvertices. By using the results in [25], we completely solved the problem for the graph parameterχ, for allnandkwithn≥2kas stated in the following theorems.

Theorem5.1 [29]. Letn,k,rbe integers such that3≤k≤r≤n−2. LetD(n,k)=max{r: χ(n,k,r)= ∅}. Supposen=kt+lwitht≥2and0≤l≤k−1. Let(n,t)=1or0 accord-ing to whetherntis odd or even. Then

(1)D(kt,k)=(k−1)t,

(2)D(kt+k−1,k)=(k−1)t+k−3−(n,t+ 1), (3)D(kt+l,k)=(k−1)t+l−1−(n,t),1≤l≤k−2.

Under the conditions ofTheorem 5.1, we get the following theorem. Theorem5.2 [29]. χ(n,k,r)= ∅ifk≤r≤D(n,k).

It is interesting to solve the problem by removing the conditionn≥2k. It is also inter-esting to solve the problem for other graph parameters such asω,φ,α0,α1, andγ.

6. Applications

In 1963, Erd˝os and Gallai [11] proved that any regular graph onnvertices has chromatic numberk≤3n/5 unless the graph is complete. Commenting on their result in a personal communication, Erd˝os wrote,“probably such a graph exists for everyk≤3n/5, except possibly for trivial exceptional cases.”

onnvertices. By using the interpolation results on graph parameterχin [25], we are able to provide an alternate proof of Erd˝os’ question.

In [24] we generalized the result of Erd˝os and Gallai by introducing the concept of F(j)-graphs and showed that their result is a special case whenj=3.

Letnbe an integer withn≥4,r=n−3 andn=5p+i, 0≤i≤4. Then an explicit formula for max(χ,rn_{) can be obtained as follows:}

maxχ,rn₌

        

3p ifi=0, 1, 3p+ 1 ifi=2, 3, 3p+ 2 ifi=4.

(6.1)

For an integern≥4, letn=5p+i, 0≤i≤4, andn=3q+t, 0≤t≤2. Ifnis even andn=2m, then

χmn_{=[2,m], χ(n−3)}n₌

q+t, 3p+

i 2

. (6.2)

Ifnis odd,n≥7, andn=2m+ 1, then

χrn_{=[3,r], χ(n−3)}n_{=q+t, 3p+}i 2

, (6.3)

whereris an even integer and eitherr=m−1 orr=m. It is easy to check that

χmn_∩χ(n−3)n_{= ∅, χr}n_∩χ(n−3)n_{= ∅. (6.4)}

Thus we have the following interpolation theorem.

Theorem6.1 [30]. (1)Ifnis even andn≥6, then there exists a connected noncomplete regular graphGof ordernwithχ(G)=kif and only ifkis an integer satisfying2≤k≤3n/5. (2)Ifnis odd andn≥7, then there exists a connected noncomplete regular graphGof ordernwithχ(G)=kif and only ifkis an integer satisfying3≤k≤3n/5.

7. Suggested problems

decycling set ofG. Since for anyv∈Sthere is a connected componentCofG−Ssuch that vis adjacent to at least two vertices ofC, there existsu∈G−Ssuch thatvu=e∈E(G) andG−eis connected. Thus for two disjoint connectedr-regular graphsGandHwith minimum decycling setsSandT ofGandH, respectively, there existu∈S,v∈G−S, x∈T,y∈H−T such thatuv=e∈E(G),xy= f ∈E(H), andG−e,H−f are con-nected. A connectedr-regular graphK=((G−e)∪(H−f)) +{ux,vy}satisfies

φ(K)≤φ(G∪H)=φ(G) +φ(H), (7.1) and the following theorem holds.

Theorem7.1. Letr≥3andnrbe even. Then

Minφ,rn₌

    

r−1 ifr+ 1≤n≤2r−1,

nr−2n+ 2 2(r−1)

ifn≥2r. (7.2)

Thus the values of Min(φ,rn_{), for all r and n, are already obtained. Moreover,} Min(φ,rn_)=min(φ,rn_).

It is clear that Max(φ,rn_)=max(φ,rn_{) for allrandnwithr+ 1≤n≤2r+ 1.}

LetGbe aK5-free graph of ordern,∆(G)=4. LetFbe a maximal induced forest of

G. We denote byc(F) the number of cycles inG−F. A pair (X,Y), whereX⊆FandY⊆ G−F, is aninterchangeable pair of vertices with respect toFif (F−X)∪Yis a forest,|(F− X)∪Y| ≥ |F|, andc((F−X)∪Y)< c(F). In general, we can define an interchangeable pair of vertices for a graphG with ∆(G)>4 as follows. LetG be a graph of ordern, ∆(G)=∆>4, andGcontains noK∆+1as its component. LetF be a maximal induced

forest ofG. We denote byk(F) the number ofK∆−1inG−F. A pair (X,Y), whereX⊆F

andY⊆G−F, is aninterchangeable pair of vertices with respect toFif (F−X)∪Y is a forest,|(F−X)∪Y| ≥ |F|, andk((F−X)∪Y)< k(F).

By using the notion of interchangeable pair of vertices, we obtained in [36] the follow-ing results.

Theorem7.2. Letnbe an even integer wheren≥12. Then

Maxφ, 3n=

        

3 8n+

1

4 ifn≡2(mod 8),

3 8n

otherwise.

(7.3)

Theorem 7.3. Let G be a connectedr-regular graph of ordern≥2r+ 2. Then φ(G)≤ n(r−2)/rfor allr≥4.

Theorem7.4. Letn=rq+t. ThenMax(φ,rn_{)=n−2qift=0,Max(φ,r}n_)=n−2q−1 ift=1, 2,Max(φ,rn_{)=n−2q−2if2t > r, andMax(φ,r}n_{)∈ {n−2q−2,n−2q−1}if} 3≤t≤r/2.

Conjecture7.5. Max(φ,rn_{)=n−2q−2if3≤t≤r/2, for allr≥6.}