An acyclic edge-coloring of a graph G is a proper edge-coloring of G such that the subgraph induced by any two color classes is acyclic. The acyclic chromatic index, χa(G), is the smallest number of colors allowing an acyclic edge-coloring of G. Clearly χa(G) ≥ ∆(G) for every graph G. Cohen, Havet, and Müller conjectured that there exists a constant M such that every planar graph with ∆(G) ≥M h...

A proper edge t-coloring of a graphG is a coloring of edges of G with colors 1, 2, . . . , t such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex x is called a spectrum of x. Any nonempty subset of consecutive integers is called an interval. A proper edge t-coloring of a graph G is interval in the vertex x if the spec...

Let G be a simple, undirected graph. We say that two edges of G are within distance 2 of each other if either they are adjacent or there is some other edge that is adjacent to both of them. A distance-2-edge-coloring of G is an assignment of colors to edges so that any two edges within distance 2 of each other have distinct colors, or equivalently, a vertex-coloring of the square of the line gr...

A k-total-coloring of a graph G is a coloring of vertex set and edge set using k colors such that no two adjacent or incident elements receive the same color. In this paper, we prove that if G is a planar graph with maximum ∆ ≥ 8 and every 6-cycle of G contains at most one chord or any chordal 6-cycles are not adjacent, then G has a (∆ + 1)-total-coloring.

A Grundy coloring of a graph G is a proper vertex coloring of G where any vertex x, colored with c(x), has a neighbor of any color 1, 2, . . . , c(x)− 1. A central graph Gc is obtained from G by adding an edge between any two non adjacent vertices in G and subdividing any edge of G once. In this note we focus on Grundy colorings of central graphs. We present some bounds related to parameters of...

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 Hamming graphs, products of complete graphs, are considered.

We investigate the distinguishing index D′(G) of a graph G as the least number d such that G has an edge-colouring with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D(G) of a graph G, which is defined for colourings of vertices. We obtain a general upper bound D′(G) ≤ ∆(G) unless G is a small cycle C3, C4 or C5. We al...

A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that is adjacent to a vertex of every other color. The b-chromatic number b(G) of a graph G is the largest k such that G admits a b-coloring with k colors. A graph G is b-chromatic edge critical if for any edge e of G, the b-chromatic number of G − e is less than the b-chromatic number of G. We ...

An incidence of G consists of a vertex and one of its incident edge in G. The incidence coloring problem is a variation of vertex coloring problem. The problem is to find the minimum number (called incidence coloring number) of colors assigned to every incidence of G so that the adjacent incidences are not assigned the same color. In this paper, we propose a linear time algorithm for incidence-...

A coloring of the vertices of a graph G is said to be distinguishing provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, χD(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring....