In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K4 has b3c vertex-disjoint triangles and the vertex independence number b3c. Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are 3, 3, 79 respectively.

The chromatic symmetric function XG of a graph G was introduced by Stanley. In this paper we introduce a quasisymmetric generalization X G called the k-chromatic quasisymmetric function of G and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of XG to χG(λ), the chromatic polynomial, we also define a generalization χ k G(λ) and sh...

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G∗ be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G∗ has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G G′ has game chromatic number at most k(k+m − 1). As a consequence, the Cartesian product of two forests has game c...

We consider the problem of efficient coloring of the edges of a so-called binomial tree T, i.e. acyclic graph containing two kinds of edges: those which must have a single color and those which are to be colored with L consecutive colors, where L is an arbitrary integer greater than 1. We give an O(n) time algorithm for optimal coloring of such a tree, where n is the number of vertices of T. Al...

A proper edge coloring of a graph G is said to be acyclic if every cycle of G receives at least three colors. The acyclic edge chromatic number of G, denoted a(G), is the least number of colors in an acyclic edge coloring of G. Alon, Sudakov and Zaks [Acyclic edge coloring of graphs, J. Graph Theory 37 (2001), 157-167] conjectured that a(G) ≤ ∆(G) + 2 holds for any graph G. In present paper, we...

This is the first of two lectures on measurable chromatic numbers given in June 2010 at the University of Barcelona. Our main result here is that acyclic locally finite analytic graphs on Polish spaces admit Baire measurable 3-colorings.

let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely n-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in...