An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of a graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F ), is defined as the maximum a(G) over all the graphs G ∈ F . In this p...

We provide a characterization of several graph parameters (the acyclic chromatic number, the arrangeability, and a sequence of parameters related to the expansion of a graph) in terms of forbidden subdivisions. Let us start with several definitions. Throughout the paper, we consider only simple undirected graphs. A graph G = sdt(G) is the t-subdivision of a graph G, if G is obtained from G by r...

Abstract. Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley’s theorem relating negative evaluations of the chromatic polynomial to acyclic orientatio...

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let mad(G) and ∆(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every graph G with ∆(G) ≥ 5 and mad(G) < 3− 2 ∆ can be avd-colored with ∆(G) + 1 col...

A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two vertices are distinguished by the set colors appearing in edges incident to each vertex. The smallest value $k$ for which $G$ admits such coloring denoted $\chi'_a(G)$. We prove that $\chi'_a(G) = 2R + 1$ most circulant graphs $C_n([1, R])$.

We study a local version of gap vertex-distinguishing edge coloring. From an edge labeling f : E(G) → {1, . . . , k} of a graph G, an induced vertex coloring c is obtained by coloring the vertices with the greatest difference between incident edge labels. The local gap chromatic number χ∆(G) is ∗ Partially funded by NSF GK-12 Transforming Experiences Grant DGE-0742434. † Partially funded by Sim...

Let G be a simple graph with vertex set {v1, v2, … , vn}. The common neighborhood graph of G, denoted by con(G), is a graph with vertex set {v1, v2, … , vn}, in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G. In this paper, we compute the common neighborhood of some composite graphs. In continue, we investigate the relation between hamiltoni...

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of products of paths and cycles are considered. We determine the acyclic chromatic numbers of three such products: grid graphs...