On resolving edge colorings in graphs

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PII. S0161171203211492 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON RESOLVING EDGE COLORINGS IN GRAPHS

VARAPORN SAENPHOLPHAT and PING ZHANG

Received 26 November 2002

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving edge chromatic number of a connected graph.

2000 Mathematics Subject Classification: 05C12, 05C15.

1. Introduction. For edgese and f in a connected graphG, thedistance d(e, f )betweeneandfis the minimum nonnegative integerafor which there exists a sequencee=e0, e1, . . . , ea=f of edges ofGsuch thateiandei+1are

adjacent fori=0,1, . . . , a−1. For an edgeeofGand a subgraphF of positive size inG, thedistancebetweeneandF is defined as

d(e, F )=mind(e, f ):f∈E(F ). (1.1)

Adecompositionof a graphGis a collection of subgraphs ofG, none of which have isolated vertices, whose edge sets provide a partition of E(G). A de-composition of G into k subgraphs is a k-decomposition. A decomposition= {G1, G2, . . . , Gk}isorderedif the ordering (G1, G2, . . . , Gk) has been imposed

on Ᏸ. For an ordered k-decomposition Ᏸ= {G1, G2, . . . , Gk} of a connected

graphGande∈E(G), theᏰ-code(or simply thecode) ofeis thek-vector

c(e)=de, G1

, de, G2

, . . . , de, Gk

. (1.2)

Hence exactly one coordinate ofc(e)is 0, namely theith coordinate if e∈ E(Gi). In [3], a decompositionᏰ is defined to be a resolving decomposition

forG if every two distinct edges ofG have distinctᏰ-codes. The minimum kfor whichGhas a resolvingk-decomposition is itsdecomposition dimension dimd(G). A resolving decomposition ofGwith dimd(G)elements is aminimum

resolving decompositionforG.

A resolving decompositionᏰ= {G1, G2, . . . , Gk}of a connected graphG is

defined in [5] to beindependent ifE(Gi)is independent for eachi(1≤i≤k)

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such that c(e)c(f ) ife and f are adjacent edges of G. The minimum k for which there is an edge coloring ofG usingkdistinct colors is called the edge chromatic number χe(G)ofG. IfᏰ= {G1, G2, . . . , Gk}is an independent

decomposition of a graphG, then by assigning colorito all edges inGifor each iwith 1≤i≤k, we obtain an edge coloring ofGusingkdistinct colors. On the other hand, ifcis an edge coloring of a connected graphG, using the colors 1,2, . . . , kfor some positive integerk, thenc(e)c(f )for adjacent edges e and f inG. Equivalently,c produces a decompositionᏰ ofE(G) into color classes (independent sets) C1, C2, . . . , Ck, where the edges ofCi are coloredi

for 1≤i≤k. Thus, for an edgeein a graphG, thek-vector

c(e)=de, C1

, de, C2

, . . . , de, Ck

(1.3)

is called thecolor code(or simply thecode)c(e)ofe. If distinct edges ofGhave distinct color codes, thencis called aresolving edge coloring(orindependent resolving decomposition) ofG in [5]. Thus a resolving edge coloring of G is an edge coloring that distinguishes all edges ofGin terms of their distances from the resulting color classes. A minimum resolving edge coloring uses a minimum number of colors, and this number is theresolving edge chromatic number χr e(G)ofG. Since every resolving edge coloring is an edge coloring

and every resolving edge coloring is a resolving decomposition, it follows that

2maxdimd(G), χe(G)≤χr e(G)≤m (1.4)

for each connected graphGof sizem≥2.

To illustrate these concepts, consider the graphG ofFigure 1.1. LetᏰ1= {G1, G2, G3}be the decomposition ofG, whereE(G1)= {v1v2, v2v5},E(G2)= {v2v3, v2v6, v3v6}, andE(G3)= {v3v4, v3v5}. SinceᏰ1is a minimum resolving

decomposition ofG, it follows that dimd(G)=3. Define an edge coloringcof Gby assigning the color 1 tov1v2andv3v5, the color 2 tov2v5andv3v6, the

color 3 tov2v3, and the color 4 tov2v6andv3v4(seeFigure 1.1(b)). Sincecis

a minimum edge coloring ofG, it follows thatχe(G)=4. However,cis not a

resolving edge coloring. To see that, letᏰ2= {C1, C2, C3, C4}be the

decomposi-tion ofGinto color classes resulting fromc, where the edges inCiare colored ibyc. ThencᏰ2(v2v5)=(1,0,1,1)=cᏰ2(v3v6). On the other hand, define an edge coloringc∗ ofGby assigning the color 1 tov1v2andv3v5, the color 2

tov2v3, the color 3 to v2v5and v3v4, the color 4 to v2v6, and the color 5

tov3v6(seeFigure 1.1(c)). LetD∗= {C1, C2, . . . , C5}be the decomposition ofG

into color classes ofc∗. Then

c∗v1v2=(0,1,1,1,2), c∗v2v3=(1,0,1,1,1),

c∗v2v5

=(1,1,0,1,2), c∗v2v6

=(1,1,1,0,1), c∗v3v4

=(1,1,0,2,1), c∗v3v5

=(0,1,1,2,1), c∗v3v6

=(1,1,1,1,0).

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G: v1 v2 v3 v4 v5

v6

(a)

c: 1

2 1

4

4 2

3

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c∗: 1

3 1

3

4 5

2

(c)

Figure1.1. A graphGwith dimd(G)=3,χe(G)=4, andχr e(G)=5.

Since theD∗-codes of the edges ofG are all distinct, it follows that c∗ is a resolving edge coloring. Moreover, G has no resolving edge coloring with 4 colors and soχr e(G)=5.

The concept of resolvability in graphs has previously appeared in [7,11,12]. Slater [11,12] introduced this concept and motivated by its application to the placement of a minimum number of sonar detecting devices in a network so that the position of every vertex in the network can be uniquely determined in terms of its distance from the set of devices. Harary and Melter [7] dis-covered these concepts independently as well. Resolving decompositions in graphs were introduced and studied in [3] and further studied in [6]. Resolv-ing decompositions with prescribed properties have been studied in [5,9,10]. Resolving concepts were studied from the point of view of graph colorings in [1,2]. We refer to [4] for graph theory notation and terminology not described here.

In [5], all nontrivial connected graphs of sizemwith resolving edge chro-matic number 3 ormare characterized. Also, bounds have been established forχr e(G)of a connected graphGin terms of its size, diameter, or girth, as

stated below.

Theorem1.1. IfGis a connected graph of sizem≥3and diameterd, then

2≤χr e(G)≤m−d+3. (1.6)

Moreover,χr e(G)=2if and only ifG=P3, andχr e(G)=m−d+3if and only

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Theorem1.2. IfGis a connected graph of sizemand girth, wherem

3, then

χr e(G)≤m−+4. (1.7)

Moreover,χr e(G)=m−+4if and only ifG=Cnfor some evenn≥4.

In this paper, we study the relationships among the resolving edge chro-matic number, edge chrochro-matic number, and decomposition dimension of a connected graph, and provide bounds for the resolving edge chromatic num-ber of a connected graph in terms of other graphical parameters inSection 2. We investigate the resolving edge colorings of trees inSection 3.

2. Bounds for resolving edge chromatic numbers. In this section, we es-tablish bounds for the resolving edge chromatic number of a connected graph in terms of (1) its order and edge chromatic number; (2) its decomposition di-mension and edge chromatic number. In order to this, we need some additional definitions and preliminary results. LetᏰbe a decomposition of a connected graphG. Then a decompositionᏰofGis called arefinementofᏰif every ele-ment inᏰis a subgraph of some element of. First, we present two lemmas, the first of which appears in [9].

Lemma2.1. Letbe a resolving decomposition of a connected graphG. Ifis a refinement ofᏰ, thenis also a resolving decomposition ofG.

Lemma2.2. LetGbe a connected graph of ordern5, letT be a spanning tree ofGwithE(T )= {e1, e2, . . . , en−1}, and letH=G−E(T ). Then the decompo-sition= {F1, F2, . . . , Fn−1, H}, whereE(Fi)= {ei}for1≤i≤n−1, is a resolving

decomposition ofG.

Proof. Leteandfbe two edges ofG. Ifeandfbelong to distinct elements

ofᏰ, thenc(e)c(f ). Thus we may assume thateandfbelong to the same

elementH inᏰ. We show thatc(e)c(f ). Lete=uv, letP be the unique u−v path in T, and let u and v be the vertices on P adjacent to u and v, respectively. Iff is adjacent to at most one ofuuand vv, then either d(e, uu)d(f , uu)or d(e, vv)d(f , vv), and soc(e)c(f ). Hence we may assume thatfis adjacent to bothuuandvv. We consider two cases according to whetheru=voruv.

Case1(u=v). Thenfis incident with the vertexu. Sincen≥5 andT is a spanning tree, there is a vertexx∈V (G)−{u, v, u}such thatxis adjacent in T with exactly one ofu, v, and u. If ux∈E(T ), then d(f , ux)=1=

2=d(e, ux); otherwise,d(e, ux)=1=2=d(f , ux)or d(e, vx)=1=2= d(f , vx)according to whetheruxorvxis an edge ofT. Soc(e)c(f ).

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We now present bounds for the resolving edge chromatic number of a con-nected graph in terms of its order and edge chromatic number.

Theorem2.3. IfGis a connected graph of ordern5, then

χe(G)≤χr e(G)≤n+χe(G)−1. (2.1)

Proof. The lower bound follows by (1.4). To verify the upper bound, letm be the size ofG. IfGis a tree of ordern, thenm=n−1. Sinceχr e(G)≤m, the

result is true for a tree. Thus we may assume thatGis a connected graph that is not a tree. LetTbe a spanning tree ofGwithE(T )= {e1, e2, . . . , en−1}. LetH=

E(G)−E(T ) be the subgraph induced byE(G)−E(T ). ThenHis a nonempty subgraph ofG. Letχe(H)=kand letH1, H2, . . . , Hkbe the decomposition ofH

into the color classes resulting from a minimum edge coloring ofH. Now let

=F1, F2, . . . , Fn−1, H,∗=F1, F2, . . . , Fn−1, H1, H2, . . . , Hk, (2.2)

whereE(Fi)= {ei}for 1≤i≤n−1. SinceᏰis a resolving decomposition ofG

byLemma 2.2andD∗ is a refinement ofᏰ, it follows byLemma 2.1thatD∗ is a resolving decomposition ofGas well. ThusᏰis a resolving independent decomposition ofG, and so

χr e(G)≤∗=n+k−1=n+χe(H)−1≤n+χe(G)−1, (2.3)

as desired.

Next, we present bounds for the resolving edge chromatic number of a con-nected graph in terms of its decomposition dimension and edge chromatic number.

Theorem2.4. For every connected graphGof order at least3,

dimd(G)≤χr e(G)≤χe(G)dimd(G). (2.4)

Proof. By (1.4), it suffices to verify the upper bound: letGbe a nontriv-ial connected graph with dimd(G)=kand χe(G)=c. Furthermore, letᏰ= {G1, G2, . . . , Gk}be a resolving decomposition ofG. IfᏰ is independent, then

Ᏸis a resolving independent decomposition ofGand soχr e(G)≤ || =k=

dimd(G) < χe(G)dimd(G)sinceχe(G)≥2. Thus we may assume thatᏰis not

independent. Without loss of generality, assume thatE(Gi)is not independent

inE(G)for 1≤i≤k1≤kandE(Gi)is independent inE(G)fork1+1≤i≤k

if k1< k. Let ci=χe(Gi)for 1≤i≤k and so 1≤ci≤χe(G). Define a

de-compositionᏰ ofGfromᏰby (1) decomposing eachGi(1≤i≤k1) intoci

color classes resulting from an edge coloring ofGi; (2) retaining eachGifor k1+1≤i≤k. Certainly, Ᏸ is an independent decomposition of G with at

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ofLemma 2.1thatᏰis also an independent resolving decomposition ofG. Therefore,χr e(G)≤ || ≤ck=χe(G)dimd(G).

3. On resolving edge chromatic numbers of trees. The decomposition di-mension of a treeT was studied in [3, 6]. It was shown in [3] that Pnis the

only connected graph of ordernwith decomposition dimension 2. Although there is no general formula for the decomposition dimension of a nonpath tree, several bounds have been established for dimd(T )for such trees in [3,6].

In this section, we investigate the resolving edge chromatic number of trees. Sinceχr e(P3)=2 andχr e(Pn)=3 forn≥4, we consider trees that are not

paths. First, we need some additional definitions and notation.

A vertex of degree at least 3 in a graphGis called amajor vertex. An end-vertexuofGis said to be aterminal vertex of a major vertexvofGifd(u, v) < d(u, w)for every other major vertexwofG. Theterminal degreeter(v)of a major vertexv is the number of terminal vertices ofv. A major vertexv of Gis anexterior major vertexofGif it has positive terminal degree. Letσ (G) denote the sum of the terminal degrees of the major vertices of G and let ex(G)denote the number of exterior major vertices ofG. In fact,σ (G)is the number of end-vertices ofG. For an ordered setW= {e1, e2, . . . , ek}of edges in

a connected graphGand an edgeeofG, let

cW(e)=de, e1, de, e2, . . . , de, ek. (3.1)

The following two results are useful to us, the first of which appeared in [9] and the second of which is due to König [8].

Lemma3.1. LetT be a tree that is not a path, having ordern4andp exterior major verticesv1, v2, . . . , vp. For1≤i≤p, letui1, ui2, . . . , uiki be the

terminal vertices ofvi, letPij be thevi−uij path(1≤j≤ki), and letxij be a

vertex inPij that is adjacent tovi. Let

W=vixij: 1≤i≤p, 2≤j≤ki. (3.2)

ThencW(e)cW(f )for each paire,f of distinct edges ofT that are not edges

ofPijfor1≤i≤pand2≤j≤ki.

König’s theorem. IfGis a bipartite graph, thenχe(G)=(G). In

particu-lar, ifTis a tree, thenχe(T )=(T ).

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We are now prepared to present an upper bound for the resolving edge chromatic number of a tree that is not a path.

Theorem3.2. LetT be a tree that is not a path, having ordern4andp exterior major verticesv1, v2, . . . , vp. For1≤i≤p, letui1, ui2, . . . , uiki be the

terminal vertices ofvi, letPij be thevi−uij path(1≤j≤ki), and letxij be a

vertex inPij that is adjacent tovi. LetWbe the set described in (3.2). Then

χr e(T )≤(T−W )+σ (T )−ex(T ). (3.3)

Proof. LetU= {v1, u11, u21, . . . , up1}and letT0be the subtree ofTof small-est size that containsU. For each pair i, j of integers with 1≤i≤p and 1≤j≤ki, letQij=Pij−vibe thexij−uijpath inT. ThusT−Wis the union

of the treeT0and the pathsQij for alli,jwith 1≤i≤pand 2≤j≤ki. Since T−Wis a forest, it follows by König’s theorem thatχe(T−W )=(T−W ). We

define an edge coloringcofTby assigning (1) the colors to the edges inT−W from the set{1,2, . . . ,(T−W )}; (2) the color

cij=(T−W )+

k1+k2+···+ki−1−(i−1)

+(j−1) (3.4)

to the edge vixij inW for all i, j with 1≤i≤p and 2≤j ≤ki. Thus the

maximum color assigned to the vertices ofGbycis

cp,kp=c

vpxp,kp

=(T−W )+k1+k2+···+kp−1−(p−1)+kp−1

=(T−W )+k1+k2+···+kp−p =(T−W )+σ (T )−ex(T ).

(3.5)

Certainly, adjacent edges are colored differently bycand socis an edge col-oring ofT. It remains to show thatcis a resolving edge coloring ofT. Let

k=(T−W )+σ (T )−ex(T ) (3.6)

and letᏰ= {C1, C2, . . . , Ck}be the decomposition ofGinto the color classes

resulting from c. Since all edges inW are colored differently, it suffices to show that ife, f∈E(T−W ), thenc(e)c(f ). We consider three cases.

Case1(e, f E(T0)). ByLemma 3.1, it follows thatcW(e)cW(f ), which implies thatc(e)c(f ).

Case2(e, fE(T0)). There are two subcases.

Subcase2.1(e, fE(Qij)for somei,jwith 1ipand 2jki). Since

vixij∈W and d(e, vixij)d(f , vixij), this implies thatcW(e)cW(f ) and

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Subcase2.2(eE(Qij)and f E(Qr s), where 1i, r p, 2j, and

s≤ki). Notice that ifi=r, thenjs. Again,vixij, vrxr s∈W. Ifd(e, vixij)d(f , vixij), thenc(e)c(f ). On the other hand, ifd(e, vixij)=d(f , vixij),

thend(f , vrxr s) < d(e, vrxr s), implying thatc(e)c(f ).

Case3(exactly one ofeandf belongs toT0, sayf∈E(T0)ande∈E(Qij) for somei,jwith 1≤i≤pand 2≤j≤ki). If there is an edgew∈W such

thatf lies on thee−w path, thend(f , w) < d(e, w)and so c(e)c(f ).

Thus we may assume that every path betweeneand any edgew∈Wdoes not containf. Thenf lies on some pathP1inT for somewith 1≤≤p. We

consider two subcases.

Subcase3.1(i=). Ifd(e, vixij)d(f , vixij), thenc(e)c(f ). Thus we may assume thatd(e, vixij)=d(f , vixij). Sinceviis an exterior vertex ofT, it

follows that degvi≥3 and so there exists a branchBatvithat does not contain vixij. Necessarily,Bmust contain an edgewofW. Thend(f , w) < d(e, w)and

soc(e)c(f ).

Subcase3.2(i). Sinceviandv are exterior major vertices, it follows that degvi≥3 and degv≥3. Thus there exists a branchB1atvi that does

not containvixij and a branchB2atv that does not containvx1.

Neces-sarily, each ofB1 andB2 must contain an edge ofW. Let w1 andw2be two

edges ofT such thatwibelongs toBifori=1,2. Ifd(e, w2)d(f , w2), then cW(e)cW(f ) and so c(e)c(f ). Thus we may assume that d(e, w2)= d(f , w2). However, then,d(e, w1) < d(f , w1), implying thatcW(e)cW(f )

and soc(e)c(f ).

Thus, in any case,c(e)c(f )and soᏰis a resolving edge coloring ofG.

Therefore,χr e(T )≤(T−W )+σ (T )−ex(T ).

The upper bound in Theorem 3.2is sharp. To see this, letK1,n, n≥3, be

the star withV (K1,n)= {v, v1, v2, . . . , vn}, wherevis the central vertex ofK1,n,

and letTbe the tree obtained fromK1,nby subdividing each edgevviintovxi

andxivifor 2≤i≤n. LetW= {vxi: 2≤i≤n}. Then it can be verified that χr e(T )=(T−W )+σ (T )−ex(T )=n.

Next, we present another upper bound forχr e(T )in terms of the maximum

degree of a treeT. A major vertex of a treeT is asuperior major vertexofT if its terminal degree is at least 2. Let sup(T )denote the number of superior major vertices ofT. Thus every superior major vertex ofT is also an exterior major vertex. Hence, ifT is a tree that is not a path, then 1sup(T )≤ex(T ).

Theorem3.3. IfT is a tree that is not a path, then

χr e(T )≤(T )+sup(T ). (3.7)

Proof. Suppose thatTcontainsq1 superior major verticesv1, v2, . . . , vq. For 1≤i≤q, letui1, ui2, . . . , uiki be the terminal vertices ofvi, whereki≥2.

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letxij be the vertex inPij that is adjacent tovi, and letQij=Pij−vibe the xij−uij path inT. Furthermore, let

W∗=vixi2: 1≤i≤q (3.8)

and letT1be the subgraph ofT obtained by removing all vertices in each set V (Qij)−{xij}fromT for alli,jwith 1≤i≤qand 1≤j≤ki; that is,

T1=T−∪VQij−xij: 1≤i≤q,1≤j≤ki. (3.9)

LetQbe the linear forest whose components are the pathsQij (1≤i≤qand

1≤j≤ki) inT; that is,

Q= ∪Qij: 1≤i≤q, 1≤j≤ki

. (3.10)

Let

T0=T1−xi2: 1≤i≤q. (3.11)

ThenE(T0)=E(T1)−W∗and

E(T )=ET0

∪W∗∪E(Q). (3.12)

HenceE(T )is partitioned intoE(T0),W∗, andE(Q). We define an edge coloring c ofT by coloring the edges in each of the setsE(T0), W∗, andE(Q)in the

following three steps:

(1) ifThas only one exterior major vertex, then this exterior major vertex is also a superior major vertex sinceTis not a path. Thus∆(T0)=(T )−1

and soχe(T0)=(T )−1. Letc1be an edge coloring ofT0using∆(T )−1

colors and definec(e)=c1(e)for alle∈E(T0). IfT has more than one

exterior major vertex, then∆(T0)≤(T )and soχe(T0)≤(T ). Letc1

be an edge coloring ofT0using∆(T )colors and definec(e)=c1(e)for

alle∈E(T0);

(2) definec(vixi2)=(T )+ifor each edgevixi2inW∗, where 1≤i≤q;

(3) definec(e)for each edgeeinQ. For each pairi, jwith 1≤i≤q and 1≤j≤ki, letmij= |E(Qij)|and

EQij

= e1ij, e2ij, . . . , eijmij, (3.13)

where (1)e1

ij is incident withxij, (2)e mij

ij is incident withuij, (3)eijs is

adjacent toesij+1inQij for allswith 1≤s≤mij−1. Let

T0∗=T1

xij: 1≤i≤q, 1≤j≤ki

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For eachiwith 1≤i≤q, letdi=degT0∗vi, and so the degree ofviinT is

degvi=di+ki≤(T ). (3.15)

We consider two cases according to whetherdi=0 ordi>0.

Case1(di=0). ThusNT

0(vi)= ∅. This implies thatT has only one ex-terior major vertex that is also a superior major vertex. Notice that ifj1, j2 {1,3,4, . . . , k1}andj1≠j2, thenv1x1j1andv1x1j2are adjacent edges inT0and soc(v1x1j1)c(v1x1j2). There are two subcases.

Subcase1.1(k1=3). Define

ces 11

=cv1x13 ifsis odd, 1≤s≤m11, (3.16)

ces 11

=cv1x11 ifsis even, 2≤s≤m11, (3.17)

ces12

=(T ) ifsis odd, 1≤s≤m12, ces12

=cv1x11

ifsis even, 2≤s≤m12,

(3.18)

ces13

=(T ) ifsis odd, 1≤s≤m13, ces

13

=cv1x13 ifsis even, 2≤s≤m13.

(3.19)

Subcase1.2(k14). Fors is even and 2sm11, define c(es11)as in (3.17); for 1≤s≤m12, definec(es12)as in (3.18); for 1≤s≤m13, definec(es13)

as in (3.19). Furthermore, define

ce11s

=cv1x1k1

ifsis odd, 1≤s≤m11,

ces1j

=cv1x1,j1

ifsis odd, 1≤s≤m1j, 4≤j≤k1,

ces 1j

=cv1x1j ifsis even, 2≤s≤m1j, 4≤j≤k1.

(3.20)

Case2(di>0). Thus NT

0(vi). Letx∈NT0∗(vi). Thenvixand vixij (1≤j≤k1) are adjacent edges inT0 and so all colorsc(vix)and c(vixij),

1≤j≤k1, are distinct. There are three subcases.

Subcase2.1(ki=2). Define

cesi1=cvix

ifsis odd, 1≤s≤mi1, (3.21)

cesi1=cvixi1

ifsis even, 2≤s≤mi1, (3.22)

cesi2

=cvix

ifsis odd, 1≤s≤mi2,

ces i2

=cvixi1 ifsis even, 2≤s≤mi2.

(3.23)

Subcase2.2(ki=3). Forsis even and 2smi1, definec(es

i1)as in (3.22);

for 1≤s≤mi2, definec(esi2)as in (3.23), and define

cesi1=cvixi3

ifsis odd, 1≤s≤mi1, (3.24)

cesi3=cvix

ifsis odd, 1≤s≤mi3,

cesi3

=cvixi3

ifsis even, 2≤s≤mi3.

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Subcase2.3(ki4). Forsis even and 2smi1, definec(es

i1)as in (3.22);

for 1≤s≤mi2, definec(ei2s )as in (3.23); for 1≤s≤mi3, definec(esi3)as in

(3.25). Furthermore, define

ces i1

=cvixiki

ifsis odd, 1≤s≤mi1,

cesij=cvixi,j−1

ifsis odd, 1≤s≤mij, 4≤j≤ki,

cesij=cvixij

ifsis even, 2≤s≤mij, 4≤j≤ki.

(3.26)

Since adjacent edges ofT are colored differently byc, it follows thatcis an edge coloring ofTusing∆(T )+qcolors. It remains to show thatcis a resolving edge coloring ofT. LetᏰ= {C1, C2, . . . , C(T )+q}be the decomposition ofTinto the color classes of c. Since all edges in W∗ are colored differently byc, it suffices to show that ife, f∈E(T−W∗), thenc(e)c(f ). We consider two

cases.

Case1(there is some exterior major vertexzofT and a terminal vertexx ofzsuch thatelies on thez−xpath ofT). Letybe a vertex in thezxpath that is adjacent toz. There are two subcases.

Subcase1(a)(yz∈W). First, assume thatf lies on somez−x∗path ofT for some terminal vertexx∗ofz. Ifx=x∗, then eitherd(e, yz) < d(f , yz)or d(f , yz) < d(e, yz), implying thatc(e)c(f ). Thus we may assume that xx∗. Ifd(e, yz)d(f , yz), thenc(e)c(f ). Ifd(e, yz)=d(f , yz), then c(e)c(f )by the definition ofcand soc(e)c(f ).

Next, assume thatf does not lie on anyz−x∗ path ofT for all terminal verticesx∗ofz. If there is an edgew∈W∗such that eitherflies on thee−w path orelies on thef−wpath, thend(f , w) < d(e, w)ord(e, w) < d(f , w), respectively. In either case,c(e)c(f ). Thus, we may assume that every path betweeneand an edge ofW∗does not containfand every path betweenf and an edge ofW∗does not containe. Necessarily, then, there exist an exterior major vertexzand a terminal vertexxofzsuch thatf lies on thez−x path ofT. Sincefdoes not lie on anyz−x∗path ofTfor all terminal vertices x∗ofz, it follows thatz=z. Sincezis an exterior major vertex ofT, it follows that the degree ofzis at least 3 and so there exists a branchBatzthat does not containf. Necessarily,Bmust contain an edge ofW∗. Letw∗be an edge ofW∗that belongs toB. Ifd(e, yz)d(f , yz), thenc(e)c(f ). Thus we

may assume thatd(e, yz)=d(f , yz). This implies thatd(f , w∗) < d(e, w∗) and soc(e)c(f ).

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assume thatzz. Since the degrees ofzand zare at least 3, there exists a branchB1atzthat does not containeand a branchB2atzthat does not

containf. Necessarily,B1must contain an edgew1ofW∗and B2must

con-tain an edgew2 of W∗. If d(e, w1)d(f , w1), then c(e)c(f ), while if d(e, w1)=d(f , w1), thend(f , w2) < d(e, w2)and soc(e)c(f ).

Case2(for every exterior major vertexzofTand every terminal vertexxof

z,edoes not lie on thez−xpath ofT). Then there are at least two branches at

e, sayB1andB2, each of which contains some superior major vertex. Therefore,

each ofB1andB2contains an edge ofW∗. Letw1andw2be the edges ofW∗ inB1andB2, respectively. First assume thatf∈E(B1). Then thef−w2path of T containse, sod(e, w2) < d(f , w2)andc(e)c(f ). We now assume that f∈E(B1). Then thef−w1path ofTcontainse. Henced(e, w1) < d(f , w1), so c(e)c(f ).

Therefore,Ᏸis a resolving edge coloring ofT and soχr e(T )≤ || =(T )+

sup(T ), as desired.

In the proof ofTheorem 3.3, ifTis a tree with sup(T )≥2 such that degv≤

(T )−1 for every major vertexv ofT that is not a superior major vertex, then∆(T0)≤(T )−1. Henceχe(T0)≤(T )−1. Thus,T0has an edge coloring c∗using∆(T )−1 colors. Define an edge coloringcsuch thatc(e)=c∗(e)for alle∈E(T0)and define c(e)for eache∈V (T )−E(T0)as described in the

proof ofTheorem 3.3. Then an argument similar to the one used in the proof ofTheorem 3.3shows thatcis a resolving edge coloring ofT. Thus, we have the following corollary.

Corollary3.4. LetT be a tree withsup(T )2. If every major vertexvof

T that is not a superior major vertex hasdegv <(T ), then

χr e(T )≤(T )+sup(T )−1. (3.27)

The upper bound inCorollary 3.4is sharp. To see this, letTbe a tree having two superior major verticesv1andv2with degv1=degv2=(T )and degv <

(T )for every major vertex v ofT that is not a superior major vertex. By

Corollary 3.4,χr e(T )≤(T )+sup(T )−1=(T )+1. Assume, to the contrary,

thatχr e(T )=(T ). Letcbe a resolving edge coloring ofT with∆(T )colors

and letᏰ= {C1, C2, . . . , C(T )}be the decomposition ofT into the color classes ofc. LetN(vi)= {xi1, xi2, . . . , xi(T )}fori=1,2. Without loss of generality, assume thatxij∈Cjfori=1,2 and 1≤j≤(T ). However, then,c(v1x11)= (0,1,1, . . .)=c(v2x21), which is a contradiction. Therefore,χr e(T )=(T )+

1=(T )+sup(T )−1.

Acknowledgments. We are grateful to Professor Gary Chartrand for

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useful conversation. This research was supported in part by a Western Michi-gan University Faculty Research and Creative Activities Fund.

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[8] D. König,Über Graphen und ihre anwendung auf Determinantentheorie und Men-genlehre, Math. Ann.77(1916), 453–465 (German).

[9] V. Saenpholphat and P. Zhang,Connected resolving decompositions in graphs, to appear in Math. Bohem.

[10] ,On connected resolving decompositions in graphs, to appear in Czechoslo-vak Math. J.

[11] P. J. Slater,Leaves of trees, Proceedings of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla, 1975), Congressus Numerantium, vol. 14, Utilitas Mathematica, Manitoba, 1975, pp. 549–559.

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Varaporn Saenpholphat: Mathematics Department, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, Thailand

E-mail address:varaporn@swu.ac.th

Ping Zhang: Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA

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