Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders

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Volume 2008, Article ID 468583,11pages doi:10.1155/2008/468583

Research Article

Bipartite Diametrical Graphs of Diameter 4 and

Extreme Orders

Salah Al-Addasi1and Hasan Al-Ezeh2

1Mathematics Department, Faculty of Science, Hashemite University, Zarqa 150459, Jordan

2Mathematics Department, Faculty of Science, University of Jordan, Amman 11942, Jordan

Correspondence should be addressed to Salah Al-Addasi,salah@hu.edu.jo

Received 9 July 2007; Accepted 5 December 2007

Recommended by Peter Johnson

We provide a process to extend any bipartite diametrical graph of diameter 4 to anS-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite setsU

andW, where 2m|U| ≤ |W|, we prove that 2mis a sharp upper bound of|W|and construct an

S-graphG2m,2min which this upper bound is attained, this graph can be viewed as a

general-ization of the Rhombic Dodecahedron. Then we show that for anym ≥2, the graphG2m,2mis

the uniqueup to isomorphismbipartite diametrical graph of diameter 4 and partite sets of cardi-nalities 2mand 2m, and hence in particular, form3, the graphG6,8which is just the Rhombic

Dodecahedron is the uniqueup to isomorphismbipartite diametrical graph of such a diameter and cardinalities of partite sets. Thus we complete a characterization ofS-graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6. We prove that the neighborhoods of vertices of the larger partite set ofG2m,2mform a matroid whose basis graph is the hypercubeQ

m. We prove

that anyS-graph of diameter 4 is bipartite self complementary, thus in particularG2m,2m. Finally,

we study some additional properties ofG2m,2mconcerning the order of its automorphism group,

girth, domination number, and when being Eulerian.

Copyrightq2008 S. Al-Addasi and H. Al-Ezeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

There are frequent occasions in which graphs with a lot of symmetries are required. One such family of graphs is that ofS-graphs.S-graphs form a special type of symmetric diametrical graphs which are one of the most interesting classes of diametrical graphs, see1,2, as well as ofk-pairable graphs introduced in3. Diametrical graphs were studied under different names, see2,4–6.

All graphs we consider are assumed to be finite, connected, and to have no loops or multiple edges. For undefined notions and terminology, we use7.

Two vertices, uand v, of a nontrivial connected graphG are said to be diametrical if

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Ghas a unique diametrical vertexv, the vertex vis called the buddy ofv, see4and5. A diametrical graphGis called minimal if removing any edge ofGproduces a graph which is not diametrical of the same diameter. A diametrical graphGin whichu vEG,whenever

uvEG,is called harmonic. A diametrical graphGis called maximal if there is no diametrical graphH with the same vertex set and diameter asGsuch thatEHcontainsEGproperly. A diametrical graphGis called symmetric ifdu, v dv, u diamGfor allu, vVG, see 2, that is, a diametrical graphGis symmetric if and only ifVG Iu, ufor anyuVG, whereIu, udenotes the interval ofGconsisting of all vertices on shortestu, u-paths, see5. A symmetric diametrical graph is called an antipodal graph, and a bipartite antipodal graph is called anS-graph, see1.

It is shown in 8 that the concepts of symmetric and maximal are equivalent in the class of bipartite diametrical graphs of diameter 4. We provide a process to extend any given bipartite diametrical graph of diameter 4 to a maximal diametrical graph and hence anS-graph of the same diameter and partite sets.

For a bipartite diametrical graph of diameter 4 and partite setsUandWwith|U| ≤ |W|, it is given in8that|W| ≤1/|U2||U|. In this paper, we improve this upper bound by proving that|W| ≤ 2m, wherem 1/2|U|. To show that 2mis a sharp upper bound, we construct a

bipartite diametrical graph, which we denote byG2m,2m, of diameter 4 in which this

max-imum value of|W|is attained. In particular, whenm 3, the graphG2m,2mis indeed just

the Rhombic Dodecahedron which is sometimes also called the Rhomboidal Dodecahedron, see9. The Rhombic Dodecahedron appears in some applications in natural sciences, see, for example,10,11. It is the convex polyhedron depicted inFigure 1and can be built up by plac-ing six cubes on the faces of a seventh, takplac-ing the centers of outer cubes and the vertices of the inner as its vertices, and joining the center of each outer cube to the closer four vertices of the inner. The number|U|is a sharp lower bound of|W|which is previously known through a graph constructed in8, which we denote in this paper byG2m,2m.Form3, it is proven in12that, up to isomorphism, the graphG2m,2m KC6is the unique symmetric di-ametrical graph of order 12 and diameter 4. For largerm,G2m,2mneeds not be the unique symmetric diametrical bipartite graph of diameter 4. We give an example of anS-graph with the same cardinalities of partite sets as those ofG8,8but not isomorphic toG8,8.On the other hand, we prove that, for anym≥ 2, the graphG2m,2mis not only the uniqueup to

isomorphismS-graph of diameter 4 and partite sets of cardinalities 2mand 2m, but also is

the unique bipartite diametrical graph of such a diameter and partite sets. Consequently, the Rhombic Dodecahedron is the unique diametrical graph of diameter 4 and maximum order where the smaller partite set has cardinality 6. Joining this to the main result of12, our result about the upper bound of the cardinality of the larger partite set completes a characterization ofS-graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6.

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u2

w4

u3 w1

w3

u1 w2

w2 u1 w

3 w1

u3

w4

u2

Figure 1: The Rhombic Dodecahedron.

the vertices of the larger partite set ofG2m,2mform a matroid whose basis graph is the

hy-percubeQm. The hypercubes together with the Rhombic Dodecahedra appear as the cells in

the tight span of a totally split-decomposable metric which is used in the field of phylogenetic analysis, see14.

A bipartite graphGwith partite setsUandW is called bipartite self complementary ifG

is a subgraph ofK|U|,|W| which is isomorphic to its complementK|U|,|W|−EGwith respect to the complete bipartite graph with the same partite sets, see15. We show that the graphs

G2m,2mandG2m,2mare bipartite self complementary and study some of their properties

concerning girth, domination number, order of automorphism group, and when being Eule-rian.

2. A sharp upper bound of the cardinality of the larger partite set

One basic property of any bipartite diametrical graph of even diameter is that each of the partite sets has an even number of elements, as stated inLemma 2.1, see8.

Lemma 2.1. IfGis a bipartite diametrical graph of even diameterdand partite setsUandW,then for eachvV,bothvandvbelong to the same partite set, and hence|U|and|W|are even.

In particular, when diamG 4, each partite set ofGconsists of a vertexv, its buddyv,

and those vertices equidistant tovandv.

It is shown in2that symmetric diametrical graphs form a proper special class of max-imal diametrical graphs. But the two concepts coincide in the class of bipartite diametrical graphs of diameter 4 according to the following fact which was proven in8.

Lemma 2.2. LetGbe a bipartite diametrical graph of diameter 4. Then the following are equivalent: iGis maximal in the class of bipartite diametrical graphs with the same partite sets asG; iiGis maximal;

iiiGis symmetric.

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which is maximal in the class of bipartite diametrical graphs with the same partite sets, and hence, by the previous lemma, is anS-graph.

Theorem 2.3. LetGbe a bipartite diametrical graph of diameter 4 and partite setsUandW.Then, the following hold:

iifuU andwWsuch that the set{u, u, w, w}of vertices is independent, then the graph

Guwis bipartite diametrical of the same diameter and partite sets asG;

iiifuwEGbutu w /EG,then the graphGu wis bipartite diametrical of the same diameter and partite sets asG;

iiiif H is a graph obtained from G by repeated applications of (i) and (ii) until none of them is applicable, thenHis anS-graph of the same diameter and partite sets asG.

Proof. iAssume to the contrary thatGuw is either not diametrical or has diameter less than 4.Then there exists a vertexvVGsuch thatdGuwv, v < 4. But, sinceGuwis

also bipartite with the same partite sets asGand bothv , vbelong to the same part ofVG, thendGuwv, vis even. HencedGuwv, v 2. Thus there exists av, v-path P inGuw

containing the new edgeuwand having length 2. Then either{v, v}{u, u}andPisu w uor

{v, v}{w, w}andPisw u w, which gives a contradiction in both cases sincewu, wu /EG uw. Therefore,Guwis a diametrical graph of diameter 4.

ii LetuwEGsuch that u w /EGwithuU andwW, where U and W

are the partite sets ofG. Thenu w /EGbecause otherwise dGu, u 2, which is not the

case. Similarly,u w /EG. Now, we will show by contradiction thatGu w is diametrical of diameter 4. Assume not, then there exists a vertex vVGsuch that dGu wv, v < 4.

Clearly,Gu wis bipartite with the same partite sets asGsinceuUandwW. But both

vandvbelong to the same part ofVG, hencedGu wv, vis even, and sodGu wv, v 2.

Thus, there exists av, v-pathPinGu wcontaining the new edgeu wand having length 2. Consequently, either{v, v} {u, u}andP isu w uor{v, v} {w, w}andP isw u w, which gives a contradiction in both cases sinceuw, uw /EGu w. Therefore,Gu wis bipartite diametrical of diameter 4.

iiiLetHbe a graph obtained fromGby addition of edges according toioriiuntil none of them can be applied. ByLemma 2.2, it would be enough to show thatHis maximal in the class of bipartite diametrical graphs of diameter 4 and partite setsUandW.Assume to the contrary that there exist two nonadjacent verticesuUandwWofHsuch thatHuwstill diametrical of diameter 4.Then, sinceiiis no more applicable onH,we haveu w /EH.

But alsoiis not applicable onH,so the set{u, u, w, w}is not independent, therefore either

uwEHoruwEH.Consequently, eitherdHw, w 2 ordHu, u 2,a contradiction.

The only bipartite diametrical graph of diameter 4 in which the smaller partite set has cardinality less than 6 is the 8-cycle asLemma 2.4proven in8says.

Lemma 2.4. IfGis a bipartite diametrical graph of diameter 4 and partite setsUandWwith|U| ≤ |W|, then|U| ≥6 unlessGC8.

For a bipartite diametrical graph of diameter 4 and partite setsUandWwith|U| ≤ |W|,

we have|U| / 2, and whenever|U| 4, we must have|W| 4. It was shown in8that if

|U| ≥ 6, then|W| ≤ 2m

m,where m 1/2|U|. Thus for|U| 6, we have|W| ≤ 20. This is

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Lemma 2.5. Let u and v be distinct vertices of a diametrical graph G of diameter d3. Then

Nu/Nv.

Proof. Ifu v, thenNuNv φsincedu, v ≥ 3. Suppose thatu / v and assume to the contrary thatNuNv. Any shortest path joiningvandumust include a vertex from

Nu; letxbe such a vertex in such a path. Thendv, u dv, x dx, u dv, x 1.But

xNuNvandv is the only vertex ofGat distancedfrom v, hencedv, x d−1. Therefore,dv, u d−1 1d,contradicting the uniqueness of the buddy vertex of v.

In particular, two distinct vertices in a diametrical graph of diameter at least 3 cannot have the same neighborhood.

Before tending to give an upper bound of|W|, let us recall the following fact proven in 8.

Lemma 2.6. Every vertexvof anS-graphGof diameter 4 is adjacent to exactly half of the vertices in the partite set ofGnot containingv.

We are now in a position to prove our improvement of the upper bound of|W|.

Theorem 2.7. LetGbe a bipartite diametrical graph of diameter 4 and partite sets UandW, where 2m|U| ≤ |W|. Then,|W| ≤2m.

Proof. ByTheorem 2.3, there exists anS-graphHwith the same diameter and partite sets asG

which hasGas a subgraph. Then, byLemma 2.6, each vertexwWhas exactlymneighbors inU. Clearly,NHwcannot contain a vertex and its buddy. ButLemma 2.5implies that the |W|setsNHw,wWare mutually distinct. Hence,|W|must be less than or equal to the

number of ways of choosing single representatives from each ofmtwo-element sets; clearly that number is 2m.

Now, if G is a bipartite diametrical graph of diameter 4 and partite sets U and

W, where |U| ≤ |W|, then by Lemma 2.1, |U| 2m for some positive integer m. Since diamG 4, we must have m ≥ 2. In the next section, we will construct for each integer

m≥2 a bipartite diametrical graph of diameter 4 and partite setsUandWwith|U|2mand

|W|2m, which is the maximum possible cardinality ofWaccording to the previous theorem.

Consequently, the upper bound of|W|given inTheorem 2.7is indeed sharp.

3. Constructing the graphG2m,2mwith maximum cardinality of

the larger partite set

For a bipartite diametrical graph of diameter 4 and partite setsUandWwith 2m |U| ≤ |W|, an obvious lower bound of|W|is 2m. AnS-graph of diameter 4 and partite sets with equal cardinalities was constructed in8as follows.

Lemma 3.1. For any integerm2, the graphGwhose vertex set is the disjoint unionUW, where

U{u1, u1, u2, u2, . . . , um, um}andW{w1, w1, w2, w2, . . . , wm, wm}, and edge set is

EG uiwt: 1≤im,1≤tm1−i

uiwt: 1≤im, m1−i < tm

uiwt: 1≤im,1≤tm1−i

uiwt : 1≤im, m1−i < tm

,

3.1

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G(8, 8)

a

H

b

Figure 2: Two nonisomorphicS-graphs with minimum|W|. Any two vertices ofHat distance 2 from each other have exactly 2 common neighbors.

We will denote the graphGinLemma 3.1byG2m,2m. The graphG8,8is depicted inFigure 2a.

The next theorem provides an example of a bipartite diametrical graph of diameter 4 and partite setsUandWwith |U| ≤ |W|in which the maximum possible value of|W|given inTheorem 2.7is attained.

Theorem 3.2. Suppose thatm >1 is an integer. There is a uniqueS-graph of diameter 4 with partite setsU,Wsatisfying|U|2mand|W|2m.

Proof. Let U {u1, u1, . . . , um, um} andW {w1, . . . , w2m}. By Lemma 2.6 and the proof of

Theorem 2.7there is only one way to make anS-graph of diameter 4 with partite setsU,W

and withuiandui being buddies,i1, . . . , m: put thewjinto one-to-one correspondence with

the 2mdierent setsS{u

1, . . . , um}in whichuiis one ofui,ui for eachi1, . . . , m, and define

adjacency by settingNwj Sifwjcorresponds toSin the one-to-one correspondence. It is

straightforward to see that the graph so defined is anS-graph of diameter 4; for eachwinW,

the buddy ofwis the vertex inWwhose set of neighbors isUNw.

We will denote the graph constructed in the proof ofTheorem 3.2byG2m,2m.

The following fact follows immediately from the constructions of the graphs

G2m,2mandG2m,2m.

Corollary 3.3. For any integerm2, ifGis a bipartite diametrical graph of diameter 4 and partite setsUandW,where 2m|U| ≤ |W|, then 2m ≤ |W| ≤ 2m,where both inequalities are sharp.

Example 3.4. Form 3, the graphG2m,2m G6,8which we have already constructed is

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4. Some properties of the graphsG2m,2mandG2m,2m

We have seen in the previous section that for any integerm ≥ 2 and partite sets U andW

with 2m |U| ≤ |W|, the graphG2m,2mis a bipartite diametrical graph of diameter 4 and such partite sets with minimum possible order, whileG2m,2mis one with maximum possible

order.

Now we will show that the graph G2m,2m is always included within the graph

G2m,2m, recall that a subgraphHofGis isometric ifd

Gu, v dHu, vfor allu, vVH,

see16.

Theorem 4.1. For any integerm2, the graphG2m,2mhas an isometric subgraph isomorphic to G2m,2m.

Proof. LetU {u1, u1, u2, u2, . . . , um, um}andW {w1, w2, . . . , w2m}be the partite sets of the graphG2m,2m. Then, by the construction ofG2m,2m, for eachi 1,2, . . . , m, there exists

a unique vertexyiW such thatNyi {ui : 1 ≤ im1−i} ∪ {ui : m1−i < im}, and henceNyi {ui : 1 ≤ im1−i} ∪ {ui : m1−i < im}. Then,the set

of verticesU∪ {y1, y1, y2, y2, . . . , ym, ym}induces a subgraphH ofG2m,2misomorphic to G2m,2m. Obviously, the buddy of a vertex ofHinHis precisely its buddy inG2m,2m. To

show thatHis isometric, letxandzbe any two distinct vertices ofH. IfxzorxzEH, thendHx , z dG2m,2mx , z. So suppose thatx / zandxz /EH. Then it is clear that dHx , z dG2m,2mx , z 2 when bothxandzbelong to the same partite set ofH. Thus let xandzbe in different partite sets ofH, sayxUandzWH {y1, y1, y2, y2, . . . , ym, ym}.

Then xz /EG2m,2m for otherwise we would haveN

G2m,2mzNHz∪ {x} which contradicts the fact that degG2m,2mzm. Then,dHx , z dG2m,2mx , z 3. ThereforeHis an isometric subgraph ofG2m,2m.

The Rhombic Dodecahedron and the even cyclesC2n n ≥ 2are minimal diametrical

graphs, while the graphKC6is not, for if we remove an edgeecorresponding toK2, then the obtained graphKC6−eis again diametrical of the same diameter. Notice also that the graphKC6is justG6,6, which means thatG2m,2mneeds not be minimal. But the graph G2m,2mis minimal diametrical as the next result says.

Theorem 4.2. For any integerm2, ifHis anS-graph of diameter 4 and partite sets of cardinalities 2mand 2m,thenHis a minimal diametrical graph.

Proof. By Theorem 3.2, the graph H is isomorphic to G2m,2m. Let uw be an edge of G G2m,2m, where u U and w W. Then N

Guww NGw− {u}, NGw UNGw, and uNGw. Let xW be the vertex whose neighborhood in G is NGx NGw− {u}∪ {u}. ThenNGuwxNGuww φ. Hence,dGuww, x / 2. But dGuww, xis even, sodGuww, x≥ 4. Clearlyx / wsinceu /NGw.Therefore,xandw

are two distinct vertices withdGuww, x≥4 anddGuww, w≥4, which implies thatGuw

is not diametrical of diameter 4. Sinceuwwas an arbitrary edge ofG2m,2m,the graphH

which is isomorphic toG2m,2mis minimal.

Now we can improveTheorem 3.2and show that the graphG2m,2mis not only the

uniqueup to isomorphismS-graph of diameter 4 and partite sets of cardinalities 2mand 2m,

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Theorem 4.3. For any integerm2, ifGis a bipartite diametrical graph of diameter 4 and partite sets of cardinalities 2mand 2m, thenGis isomorphic toG2m,2m.

Proof. Let G be a bipartite diametrical graph of diameter 4 and partite sets UG and WG

with |UG| 2m and |WG| 2m. In view of Theorem 3.2, it would be enough to show

thatGis symmetric. So, assume to the contrary thatG is not symmetric. Then, by Theorem

2.3, the graphGis a proper subgraph of anS-graphH of the same diameter and partite sets asG. LetuwEHEG. ThenGHuwH, and hence for anyx, yVG,we havedGx, ydHuwx, ydHx, y. Therefore,Huwis diametrical of the same diameter

and partite sets asGandH, which contradicts the minimality ofH implied by the previous theorem.

The following characterization of the Rhombic DodecahedronG6,8is a direct conse-quence ofTheorem 4.3andTheorem 2.7.

Corollary 4.4. The Rhombic Dodecahedron is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite setsUandW,where 6|U|<|W|.

According to12, ifGis anS-graph of ordernand diameter 4 with partite setsUand

Wsuch thatG / C8, thenn≥12. Moreover, ifn12, then|U||W|6. It is also shown in 12thatKC6is the uniqueup to isomorphismsymmetric diametrical graph of order 12 and diameter 4 and hence in particular the uniqueS-graph of such an order and diameter. The previous corollary shows that the Rhombic Dodecahedron is not only the uniqueup to isomor-phismS-graph of order 14 and diameter 4, but also the uniqueup to isomorphismbipartite diametrical graph of such an order and diameter. Combining these results withLemma 2.4, we can completely characterize theS-graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6.

Corollary 4.5. LetGbe anS-graph of diameter 4 and cardinality of the smaller partite set not exceeding 6.ThenGis isomorphic to one of the graphs:C8,KC6and the Rhombic Dodecahedron.

In the set of bipartite diametrical graphs of diameter 4 and partite setsUandW with

2m|U| ≤ |W|, although the graphG2m,2mis the uniqueup to isomorphismwith

maxi-mum order, the graphG2m,2mneeds not be the only one with minimum order, for example, the graphsKC6G6,6andKC6−e, whereeis an edge correspoding toK2, both have minimum order. Even under the condition of being symmetric, althoughG6,6was proven in 12to be the only one,G2m,2min general needs not be unique, for example,G8,8and the graphHof diameter 4 inFigure 2are nonisomorphicS-graphs with minimum order.

Now we will prove that the set of all neighborhoods of vertices of the larger partite set in the graphG2m,2mforms a matroid on the smaller partite set, and determine its basis graph.

Theorem 4.6. LetUandWbe the partite sets ofG2m,2mwith cardinalities 2mand 2m, respectively.

Thenβ{Nw: wW}forms a matroid onUwhose basis graph is the hypercubeQm.

Proof. Let Nwi, Nwj be two elements ofβwith aNwiNwj. Then aNwj.

Thus the set S Nwi − {a} ∪ {a} has exactly m elements and contains no pair of

diametrical vertices. So by the construction of G2m,2m, there exists a vertex w

kW

such that Nwk S. Therefore, Nwi − {a} ∪ {a} ∈ β and hence β is a matroid.

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a baseNwiby a single exchange. ThenNwj Nwi− {a}∪ {b}for someaNwiand

someb /Nwi. But the setNwjcontains exactly one ofa, a. Thenbmust bea. Therefore,

a single exchange is precisely replacing a vertex by its buddy, which means that two vertices inare adjacent if and only if one of them can be obtained from the other by replacing a vertex fromUby its buddy. Now sinceU {u1, u1, u2, u2, . . . , um, um}, we can represent each

vertexNwofby a binary stringr1r2· · ·rm,where fori1,2, . . . , m,we put

ri

0, uiNw,

1, uiNw,

4.1

thus two vertices ofare adjacent if and only if their binary representations differ at exactly one place. This is just the binary representation of the hypercubeQm.

It is shown in8thatG2m,2m has exactly 8m automorphisms. This indicates that

G2m,2mhas a lot of symmetry. The next result reflects the higher symmetryG2m,2m

pos-sesses.

Theorem 4.7. For any integerm3, the automorphism group AutG2m,2mhas order 2mm!.

Proof. Let f be an automorphism of G2m,2m. Then, sincef preserves degrees of vertices

and|U| / |W|, it follows byLemma 2.6 that the vertexfw1 wmust belong toW. Then

fNw1 Nw,and hence sincef preserves distances between vertices, we must have

fu fufor every vertexuUNw1 Nw1. Now for any vertexxW− {w1}, by the construction ofG2m,2m, there exists a unique vertexyW such thatNy fNx

and hencefx y. This gives an automorphismf ofG2m,2m. Thus an automorphism of G2m,2mis completely defined by specifying the image of every vertex in the closed

neigh-borhoodNw1. From the proof above, the image ofw1can be any vertex fromW, and the set of images of the vertices fromNw1must only be the set of neighbors offw1. This implies that we have exactly 2mways to choosefw1and thenm! ways to choose the set of images of

the vertices fromNw1. Therefore,G2m,2mhas exactly 2mm! automorphisms.

One can easily verify that theS-graphs of diameter 4 and order at most 14, already char-acterized inCorollary 4.5, are bipartite self complementary. The following result assures that this is the case for anyS-graph of diameter 4.

Theorem 4.8. IfGis anS-graph of diameter 4, thenGis bipartite self complementary.

Proof. Let U {u1, u2, . . . , uk, u1, u2, . . . , uk}and W {w1, w2, . . . , wt, w1, w2, . . . , wt} be the

partite sets ofGwhich is a subgraph ofK2k,2t. Then, byLemma 2.6, each vertexxfromUis

adjacent to exactly half of the vertices ofW, andxis adjacent to the other half. Thus, the one-to-one correspondenceφfrom VGtoVK2k,2tEG, which fixes every vertex inW and

sends every vertex inUto its buddy, preserves adjacency.

Consequently, bothG2m,2mandG2m,2mare bipartite self complementary.

Finally, we close this section by the following result including some properties of

G2m,2mandG2m,2mconcerning girth, domination number, and being Eulerian.

Theorem 4.9. iThe girth of each of the graphsG2m,2mandG2m,2m,wherem 3,is 4; while

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iiThe domination number of anyS-graph of diameter 4 is 4. Hence, each ofG2m,2mand

G2m,2mhas domination number 4.

iiiAnyS-graph of diameter 4 is Eulerian if and only if the cardinality of each of its partite sets is divisible by 4. Hence, the graphsG2m,2mandG2m,2mare Eulerian if and only ifmis even.

Proof. i The graph G4,4 is the 8-cycle. Now consider m ≥ 3. In the graph G2m,2m,

we haveNu1 {w1, w2, . . . , wm}andNu2 {w1, w2, . . . , wm−1} ∪ {wm}. Thenu1w1u2w2u1 is a 4-cycle in the bipartite graphG2m,2mwhich implies that the girth ofG2m,2mis 4. Now sinceG2m,2mis bipartite, then byTheorem 4.1, its girth is also 4.

ii Let G be an S-graph of diameter 4 and partite sets U and W. Then by

Lemma 2.6, each vertex in one of the partite sets is adjacent to exactly half of the vertices of the other partite set. Thus,Gcannot have a dominating set with number of elements less than 4. But for any two verticesu, w,whereuUandwW, we haveNuNu W and

NwNw U. Therefore,{u, u, w, w}is a dominating set ofG.

iiiLetGbe anS-graph of diameter 4 and partite setsU andW. ByLemma 2.6, each vertex ofGhas degree either1/2|U|or1/2|W|. Therefore, the degree of every vertex ofG

is even if and only if both |U|and |W|are multiples of 4.

One might put for further research the following question. For anS-graph of diameter 4 and partite setsUandW,where 2m |U| ≤ |W|, a sharp lower bound and a sharp upper bound of|W|are 2mand 2m, respectively. Ifm >3,then for each integertwithm < t <2m−1, what about the existence of anS-graph of diameter 4 and partite setsUandW,where 2m|U|

and|W|2t?

Acknowledgment

The authors are grateful to the referees for their valuable suggestions.

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Figure

Figure 1: The Rhombic Dodecahedron.

Figure 1:

The Rhombic Dodecahedron. p.3
Figure 2: Two nonisomorphic S-graphs with minimum |W|. Any two vertices of H at distance 2 from eachother have exactly 2 common neighbors.

Figure 2:

Two nonisomorphic S-graphs with minimum |W|. Any two vertices of H at distance 2 from eachother have exactly 2 common neighbors. p.6

References

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