 Home  About us  Contact  # Edge Coloring (edge + coloring)

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## Selected Abstracts

### d -Regular graphs of acyclic chromatic index at least d+2

JOURNAL OF GRAPH THEORY, Issue 3 2010
Manu Basavaraju
Abstract An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a,(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a,(G),,+2, where ,=,(G) denotes the maximum degree of the graph. Alon et al. also raised the question whether the complete graphs of even order are the only regular graphs which require ,+2 colors to be acyclically edge colored. In this article, using a simple counting argument we observe not only that this is not true, but in fact all d -regular graphs with 2n vertices and d>n, requires at least d+ 2 colors. We also show that a,(Kn, n),n+ 2, when n is odd using a more non-trivial argument. (Here Kn, n denotes the complete bipartite graph with n vertices on each side.) This lower bound for Kn, n can be shown to be tight for some families of complete bipartite graphs and for small values of n. We also infer that for every d, n such that d,5, n,2d+ 3 and dn even, there exist d -regular graphs which require at least d+2-colors to be acyclically edge colored. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 226,230, 2010 [source]

### Multicolored trees in complete graphs

JOURNAL OF GRAPH THEORY, Issue 3 2007
S. Akbari
Abstract A multicolored tree is a tree whose edges have different colors. Brualdi and Hollingsworth 5 proved in any proper edge coloring of the complete graph K2n(n > 2) with 2n , 1 colors, there are two edge-disjoint multicolored spanning trees. In this paper we generalize this result showing that if (a1,,, ak) is a color distribution for the complete graph Kn, n,,,5, such that , then there exist two edge-disjoint multicolored spanning trees. Moreover, we prove that for any edge coloring of the complete graph Kn with the above distribution if T is a non-star multicolored spanning tree of Kn, then there exists a multicolored spanning tree T' of Kn such that T and T' are edge-disjoint. Also it is shown that if Kn, n , 6, is edge colored with k colors and , then there exist two edge-disjoint multicolored spanning trees. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 221,232, 2007 [source]

### Rainbow trees in graphs and generalized connectivity

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2010
Gary Chartrand
Abstract An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 , k , n. A k -rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S , V(T). The minimum number of colors needed in a k -rainbow coloring of G is the k -rainbow index of G. For every two integers k and n , 3 with 3 , k , n, the k -rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection {T1,T2,,,T,} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) , V(Tj) = S for every pair i,j of distinct integers with 1 , i,j , ,. For an integer k with 2 , k , n, the k -connectivity ,k(G) of G is the greatest positive integer , for which G contains at least , internally disjoint trees connecting S for every set S of k vertices of G. It is shown that ,k(Kn)=n,,k/2, for every pair k,n of integers with 2 , k , n. For a nontrivial connected graph G of order n and for integers k and , with 2 , k , n and 1 , , , ,k(G), the (k,,)-rainbow index rxk,,(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least , internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rxk,,(Kn) are determined for all possible values k and , when n , 6. It is also shown that for , , {1, 2}, rx3,,(Kn) = 3 for all n , 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source]

### Vertex-distinguishing edge colorings of random graphs

RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2002
P. N. Balister
A proper edge coloring of a simple graph G is called vertex-distinguishing if no two distinct vertices are incident to the same set of colors. We prove that the minimum number of colors required for a vertex-distinguishing coloring of a random graph of order n is almost always equal to the maximum degree ,(G) of the graph. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 89,97, 2002 [source]