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...

An adjacent vertex distinguishing edge-coloring of a graph G is a proper edge-coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing edge-coloring of G is denoted by χ a (G). We prove that χ a (G) is at most the maximum degree plus 2 if G is a planar graph without isolated edges w...

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index χspGq of a graph G is the minimum number of colors in a strong edge coloring of G. Let ∆ ě 4 be an integer. In this note, we study the properties of the odd graphs, and show that every planar graph with maximum degree at most ∆ and girth at least 10∆ ́ 4 has a...

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that ∆(G) + 2 colors suffice for an acyclic edge coloring of every graph G [6]. The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is ∆+12 [2]. In this paper, we study simple planar graph...

A μ-simultaneous edge coloring of graph G is a set of μ proper edge colorings of G with a same color set such that for each vertex, the sets of colors appearing on the edges incident to that vertex are the same in each coloring and no edge receives the same color in any two colorings. The μ-simultaneous edge coloring of bipartite graphs has a close relation with μ-way Latin trades. Mahdian et a...

A skew edge coloring of a graph G is defined to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, skew edge coloring of certain classes of graphs are determined. Furthermore, the skew chromatic index of those graphs is obtained ...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

Let G(V,E) be a graph, and let f be an integer function on V with 1 ≤ f(v) ≤ d(v) to each vertex v ∈ V . An f -edge cover coloring is an edge coloring C such that each color appears at each vertex v at least f(v) times. The f -edge cover chromatic index of G, denoted by χ ′ fc(G), is the maximum number of colors needed to f -edge cover color G. It is well known that min v∈V {bd(v)− μ(v) f(v) c ...

We study the weighted improper coloring problem, a generalization of defective coloring. We present some hardness results and in particular we show that weighted improper coloring is not fixed-parameter tractable when parameterized by pathwidth. We generalize bounds for defective coloring to weighted improper coloring and give a bound for weighted improper coloring in terms of the sum of edge w...

Vertex coloring of a graph is the assignment of labels to the vertices of the graph so that adjacent vertices have different labels. In the case of polyhedral graphs, the chromatic number is 2, 3, or 4. Edge coloring problem and face coloring problem can be converted to vertex coloring problem for appropriate polyhedral graphs. We have been developed an interactive learning system of polyhedra,...