The acyclic edge colouring problem is extensively studied in graph theory. The corner-stone of this field is a conjecture of Alon et. al.[1] that a′(G) ≤ ∆(G) + 2. In that and subsequent work, a′(G) is typically bounded in terms of ∆(G). Motivated by this we introduce a term gap(G) defined as gap(G) = a′(G) − ∆(G). Alon’s conjecture can be rephrased as gap(G) ≤ 2 for all graphs G. In [5] it was...

Let P (k) be the chromatic polynomial of a graph with n ≥ 2 vertices and no isolated vertices, and let R(k) = P (k + 1)/k(k + 1). We show that the coefficients of the polynomial (1 − t) ∑ ∞ k=1 R(k)t are nonnegative and we give a combinatorial interpretation to R(k) when k is a nonpositive integer.

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic (2colored) cycles. The acyclic chromatic index of a graph G, denoted by a(G), is the least integer k such that G admits an acyclic edge-coloring using k colors. Let ∆ = ∆(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by Kn,n. Basavaraju, C...

In the main this paper introduces the concept of chromatic harmonic polynomials denoted, $H^chi(G,x)$ and chromatic harmonic indices denoted, $H^chi(G)$ of a graph $G$. The new concept is then applied to finding explicit formula for the minimum (maximum) chromatic harmonic polynomials and the minimum (maximum) chromatic harmonic index of certain graphs. It is also applied to split graphs and ce...

Abstract Let G be a graph. We introduce the acyclic b-chromatic number of as an analogue to . An coloring graph is map $$c:V(G)\rightarrow \{1,\ldots ,k\}$$ c : V ( G ) → { 1 ...

In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P4-tidy graphs and (q, q − 4)-graphs, for every fixed q. These classes include cographs, P4-sparse and P4-lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of (q, q − 4)-graphs. All these coloring pro...

In the thesis, the coloring of digraphs is studied. The chromatic number of a digraph D is the smallest integer k so that the vertices of D can be partitioned into at most k sets each of which induces an acyclic subdigraph. A set of four topics on the chromatic number is presented. First, the dependence of the chromatic number of digraphs on the maximum degree is explored. An analog of Gallai’s...

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orienta...

We prove the theorem from the title: the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov and Zaks and presents further support for a conjecture that ∆(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analogue of a result of Robinson and the second auth...