Cost Edge-Coloring of a Cactus

Let C be a set of colors, and let ?be an integer cost assigned to a color c in C. An edge-coloring of a graph ?is assigning a color in C to each edge ?so that any two edges having end-vertex in common have different colors. The cost ?of an edge-coloring f of G is the sum of costs ?of colors ?assigned to all edges e in G. An edge-coloring f of G is optimal if ?is minimum among all edge-colorings of G. A cactus is a connected graph in which every block is either an edge or a cycle. In this paper, we give an algorithm to find an optimal edge- ??coloring of a cactus in polynomial time. In our best knowledge, this is the first polynomial-time algorithm to find an optimal edge-coloring of a cactus.

A Note on the Strong Edge-coloring of Outerplanar Graphs with Maximum Degree 3

A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, axe assigned different colors. The strong chromatic index of G is the smallest integer k for which G has a strong k-edge-coloring. In this paper, we have shown that the strong chromatic index is no larger than 6 for outerplanax graphs with maximum degree 3.

Existence of rainbow matchings in properly edge-colored graphs

Let G be a properly edge-colored graph. A rainbow matching of G is a matching in which no two edges have the same color. Let 5 denote the minimum degree of G. We show that if Iv（G）I 〉 （σ2 ＋ 14σ ＋ 1）/4, then G has a rainbow matching of size 6, which answers a question asked by G. Wang [Electron. J. Combin., 2011, 18： #N162] affirmatively. In addition, we prove that if G is a properly colored bipartite graph with bipartition （X, Y） and max{lXl, IYI} 〉 （σ2 ＋ 4σ - 4）/4, then G has a rainbow matching of size σ.

Conflict-free Connection Number and Independence Number of a Graph

An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path,which contains a color used on exactly one of its edges.The conflict-free connection number of a connected graph G,denoted by cf c(G),is defined as the minimum number of colors that are required in order to make G conflict-free connected.In this paper,we investigate the relation between the conflict-free connection number and the independence number of a graph.We firstly show that cf c(G)≤α(G)for any connected graph G,and give an example to show that the bound is sharp.With this result,we prove that if T is a tree with?(T)≥(α(T)+2)/2,then cf c(T)=?(T).

Some Class 1 Graphs on gc-colorings

An edge-coloring of a graph G is an coloring of a graph G is an edge-coloring of G such assignment of colors to all the edges of G. A go- that each color appears at each vertex at least g（v） times. The maximum integer k such that G has a go-coloring with k colors is called the gc-chromatic index of G and denoted by X＇gc （G）. In this paper, we extend a result on edge-covering coloring of Zhang X＇gc（ ） = δg（G）, and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy ＇ x＇gc（G）=δg（G）,where δg（G）=minv∈V（G）{[d（v）/g（v）]}.

f-Colorings of Some Graphs of f-Class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v V（G） at most f（v） times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by X′f（G）. Any simple graph G has the f-chromatic index equal to △f（G） or △f（G） ＋ 1, where △f（G） =max v V（G）{[d（v）/f（v）]}. If X′f（G） = △f（G）, then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with △f（G） colors are given.

Bounds for the Rainbow Disconnection Numbers of Graphs

An edge-cut of an edge-colored connected graph is called a rainbow cut if no two edges in the edge-cut are colored the same.An edge-colored graph is rainbow disconnected if for any two distinct vertices u and v of the graph,there exists a rainbow cut separating u and v.For a connected graph G,the rainbow disconnection number of G,denoted by rd(G),is defined as the smallest number of colors required to make G rainbow disconnected.In this paper,we first give some upper bounds for rd(G),and moreover,we completely characterize the graphs which meet the upper bounds of the NordhausGaddum type result obtained early by us.Secondly,we propose a conjecture that for any connected graph G,either rd(G)=λ^(+)(G)or rd(G)=λ^(+)(G)+1,whereλ^(+)(G)is the upper edge-connectivity,and prove that the conjecture holds for many classes of graphs,which supports this conjecture.Moreover,we prove that for an odd integer k,if G is a k-edge-connected k-regular graph,thenχ’(G)=k if and only if rd(G)=k.It implies that there are infinitely many k-edge-connected k-regular graphs G for which rd(G)=λ^(+)(G)for odd k,and also there are infinitely many k-edge-connected k-regular graphs G for which rd(G)=λ^(+)(G)+1 for odd k.For k=3,the result gives rise to an interesting result,which is equivalent to the famous Four-Color Problem.Finally,we give the relationship between rd(G)of a graph G and the rainbow vertex-disconnection number rvd(L(G))of the line graph L(G)of G.

BOUNDS ON THE LOWER SIZE OF A 7-CRITICAL GRAPH

In this paper, the authors have obtained some lower bounds on the size of a 7-critical graph,and some results about the planar graph conjecture have been given.

Partitioning Complete Graphs by Heterochromatic Trees

A heterochromatie tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr （G）, is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of r-edge-colored complete graphs. We also find at most tr（Kn） vertex-disjoint heterochromatic trees to cover all the vertices in polynomial time for a given r-edge-coloring of Kn.