The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

Three edges e1, e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph G = (V,E) is a coloring c of the edges of G such that if e1, e2 and e3 are consecutive edges in G, then c(e1) 6= c(e3). The injective edge coloring number χ ′ i(G) is the minimum number of colors permitted in such a coloring. In this paper,...

We determine the exact values of the circular chromatic index of the Goldberg snarks, and of a related family, the twisted Goldberg snarks.

Musical diachrony and synchrony have revealed an incredible variation in musical scales, many of which are implied, lurking in traditional patterns and not adequately transcribed in their projection to the dominant Western music predicates. From antiquity the term “chromatic” was used to determine the coordinates of diversification in terms of psychoacoustic perception of music, and it yielded ...

Corresponding to the game chromatic number of graphs, we consider in this paper the game chromatic index χg of graphs, which is defined similarly, except that edges, instead of vertices of graphs are colored. Upper bounds for trees and wheels are given.

A well known conjecture of Hadwiger asserts that Kn+1 is the only minor minimal graph of chromatic number greater than n. In this paper, all minor minimal graphs of chromatic index greater than n are determined. c © 2008 Elsevier B.V. All rights reserved.

In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the de0ciency is very small. c © 2001 Elsevier Science B.V. All rights reserved.

This paper is an expository piece on edge-chromatic graph theory. The central theorem in this subject is that of Vizing. We shall then explore the properties of graphs where Vizing’s upper bound on the chromatic index is tight, and graphs where the lower bound is tight. Finally, we will look at a few generalizations of Vizing’s Theorem, as well as some related conjectures.

We complete the determination of the chromatic number of 6valent circulants of the form C(n; a, b, a+b) and show how this can be applied to improving the upper bound on the chromatic index of cyclic Steiner triple systems.

let $g$ be a non-abelian group and let $z(g)$ be the center of $g$. associate with $g$ there is agraph $gamma_g$ as follows: take $gsetminus z(g)$ as vertices of$gamma_g$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $gamma_g$ is called the non-commuting graph of $g$. in recent years many interesting works have been done in non-commutative graph of groups. computing the clique...