A hole in a graph G $G$ is an induced cycle of length at least four, and k $k$ -multihole the union pairwise disjoint nonneighbouring holes. It well known that if does not contain any holes then its chromatic number equal to clique number. In this paper we show that, for integer ≥ 1 $k\ge 1$ , -multihole, most polynomial function We same result holds ask all be odd or four; longer than fixed co...

We introduce a notion of color-criticality in the context chromatic-choosability. define graph $G$ to be strong $k$-chromatic-choosable if $\chi(G) = k$ and every $(k-1)$-assignment for which is not list-colorable has property that lists are same all vertices. That usual coloring is, some sense, obstacle list-coloring. prove basic properties strongly chromatic-choosable graphs such as chromatic...

Collins and Trenk define the distinguishing chromatic number χD(G) of a graph G to be the minimum number of colors needed to properly color the vertices of G so that the only automorphism of G that preserves colors is the identity. They prove results about χD(G) based on the underlying graphG. In this paper we prove results that relate χD(G) to the automorphism group of G. We prove two upper bo...

The chromatic polynomial and its generalization, the symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, Stanley show how both objects ca...

The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum. A natural generalization of chromatic sum is optimum cost chromatic partition (OCCP) problem, where the costs of colors can be arbitrary positive numbers. Existing results about chromatic sum, s...

Recently, Tittmann et al. introduced the subgraph component polynomial and showed that its power for distinguishing graphs is quite different from the power of other graph polynomials that appear in the literature such as the matching polynomial, the Tutte polynomial, the characteristic polynomial, the chromatic polynomial, etc. The subgraph component polynomial enumerates vertex induced subgra...

As proved by Kahn, the chromatic number and fractional chromatic number of a line graph agree asymptotically. That is, for any line graph G we have χ(G) ≤ (1 + o(1))χf (G). We extend this result to quasi-line graphs, an important subclass of claw-free graphs. Furthermore we prove that we can construct a colouring that achieves this bound in polynomial time, giving us an asymptotic approximation...