Let H be a hypergraph. For a k-edge coloring c : E(H) → {1, . . . , k} let f(H, c) be the number of components in the subhypergraph induced by the color class with the least number of components. Let fk(H) be the maximum possible value of f(H, c) ranging over all k-edge colorings of H . If H is the complete graph Kn then, trivially, f1(Kn) = f2(Kn) = 1. In this paper we prove that for n ≥ 6, f3...

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...

The paper is devoted to the model of compact cyclic edge-coloring. This variant of edge-coloring finds its applications in modeling schedules in production systems, in which production proceeds in a cyclic way. We point out optimal colorings for some graph classes and we construct graphs which cannot be colored in a compact cyclic manner. Moreover, we prove some theoretical properties of consid...

The focus of this article is on three of the author’s open conjectures. The article itself surveys results relating to the conjectures and shows where the conjectures are known to hold.

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs.

An edge-operation on a graph G is defined to be either the deletion of an existing edge or the addition of a nonexisting edge. Given a family of graphs G, the editing distance from G to G is the smallest number of edge-operations needed to modify G into a graph from G. In this paper, we fix a graph H and consider Forb(n,H), the set of all graphs on n vertices that have no induced copy of H. We ...

An r-edge coloring of a graph G is a mapping h : E(G) → [r], where h(e) is the color assigned to edge e ∈ E(G). An exact r-edge coloring is an r-edge coloring h such that there exists an e ∈ E(G) with h(e) = i for all i ∈ [r]. Let h be an edge coloring of G. We say G is rainbow if no two edges in G are assigned the same color by h. The anti-Ramsey number, AR(G,n), is the smallest integer r such...

The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with (G) = , then the first player, Alice, has a winning strategy for this game with...

We consider lower bounds on the the vertex-distinguishing edge chromatic number of graphs and prove that these are compatible with a conjecture of Burris and Schelp [8]. We also find upper bounds on this number for certain regular graphs G of low degree and hence verify the conjecture for a reasonably large class of such graphs.