In this paper we explore structural properties of unitary Cayley graphs, including clique and chromatic number, vertex and edge connectivity, planarity, and crossing number.

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

A connected graph G with chromatic number t is double-critical if G − x − y is (t − 2)-colorable for each edge xy ∈ E(G). The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erdős and Lovász from 1966, which is referred to as the Double-critical Graph Conjecture, states that there are no other double-critical graphs, i.e., if a graph G with c...

For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same and let V 1 , V 2 ,. .. , V k be the resulting color classes. For a vertex v of G, the metric color code of v is the k-vector code(v) = (d(v, V 1), d(v, V 2), · · · , d(v, V k)), where d(v, V i) is the minimum distance between v and a vertex in V i. If code(u) = code(v...

The induced arboricity of a graph G is the smallest number of induced forests covering the edges of G. This is a well-defined parameter bounded from above by the number of edges of G when each forest in a cover consists of exactly one edge. Not all edges of a graph necessarily belong to induced forests with larger components. For k > 1, we call an edge k-valid if it is contained in an induced t...

An acyclic edge coloring of a graph G is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The acyclic chromatic index χa(G) of a graph G is the least number of colors needed in any acyclic edge coloring of G. Fiamčík (1978) conjectured that χa(G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G...

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that ...

This paper contains two principal results. The first proves that any graph G can be given a balanced proper edge coloring by k colors for any k ≥ χ′(G). Here balanced means that the number of vertices incident with any set of d colors is essentially fixed for each d, that is, for two different d-sets of colors the number of vertices incident with each of them can differ by at most 2. The second...

A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20.

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). It was conjectured by Alon, Sudakov and Zaks that for any simple and finite graph G, a(G) ≤ ∆+ 2, where ∆ = ∆(G) denotes the maximum degree of G. ...