a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...

for a graph $g$, let $p(g,lambda)$ denote the chromatic polynomial of $g$. two graphs $g$ and $h$ are chromatically equivalent if they share the same chromatic polynomial. a graph $g$ is chromatically unique if any graph chromatically equivalent to $g$ is isomorphic to $g$. a $k_4$-homeomorph is a subdivision of the complete graph $k_4$. in this paper, we determine a family of chromatically uni...

A graph with chromatic number k is called k-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 23 vertices and all triangle-free 5-chromatic graphs on 24 vertices with maximum degree at most 7. This implies that Reed’s conject...

Chromatic number, chromatic sum and chromatic sum number are important graph coloring characteristics. The paper proves that a parallel metaheuristic like the parallel genetic algorithm (PGA) can be efficiently used for computing approximate sum colorings and finding upper bounds for chromatic sums and chromatic sum numbers for hard– to–color graphs. Suboptimal sum coloring with PGA gives usual...

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...

1 The Chromatic Complex 2 1.1 The Chromatic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Quantum Dimension of Graded Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Categorification of the Chromatic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 Enhanced States . . . . . . . . . . . . . . . . . . . ....

In this paper, I give a short proof of a recent result by Sokal, showing that all zeros of the chromatic polynomial PG(q) of a finite graph G of maximal degree D lie in the disc |q| < KD, where K is a constant that is strictly smaller than 8.

In this paper we study various extremal problems related to some combinatorially defined graph polynomials such as matching polynomial, chromatic polynomial, Laplacian polynomial. It will turn out that many problems attain its extremal value in the class of threshold graphs. To attack these kinds of problems we survey several applications of the so-called Kelmans transformation. Mea culpa. This...

For any positive integers a, b, c, d, let Ra,b,c,d be the graph obtained from the complete graphs Ka,Kb,Kc and Kd by adding edges joining every vertex in Ka and Kc to every vertex in Kb and Kd. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of Ra,b,c,d is either a real number or a non-real number with its real part equal to (a + b + c...

In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete) non-uniform (m, l)− hyperwheels and non-uniform (m, l)−hyperfans. These hypergraphs, constructed through a “join” graph operation, are some generalizations of the well-known wh...