A twin edge k-coloring of a graph G is a proper edge k-coloring of G with the elements of Zk so that the induced vertex k-coloring, in which the color of a vertex v in G is the sum in Zk of the colors of the edges incident with v, is a proper vertex k-coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Twin chromatic index of the square P 2 n ...

A strong edge-coloring of a graph G is a function that assigns to each edge a color such that two edges within distance two apart must receive different colors. The minimum number of colors used in a strong edge-coloring is the strong chromatic index of G. Lih and Liu [14] proved that the strong chromatic index of a cubic Halin graph, other than two special graphs, is 6 or 7. It remains an open...

Let G = (V, E) be a graph and k > 2 an integer. The general chromatic index x;(G) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph (F) induced by F has size at most k1. This paper initiates a study of x;(G) and generalizes some known results on chromatic index. The purpose of this paper is to obtain a generalization of chromatic index....

Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic...

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 of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper we prove that if G is a graph with maximu...

We shall use the distance chromatic index defined by the present author in early nineties, cf. [5] or [4] of 1993. The edge distance of two edges in a multigraph M is defined to be their distance in the line graph L(M) of M . Given a positive integer d, define the d+-chromatic index of the multigraph M , denoted by q(d)(M), to be equal to the chromatic number χ of the dth power of the line grap...

Our motivation is the question how similar the f -colouring problem is to the classic edge-colouring problem, particularly with regard to graph parameters. In 2010, Zhang, Yu, and Liu [9] gave a new description of the f -matching polytope and derived a formula for the fractional f -chromatic index, stating that the fractional f -chromatic index equals the maximum of the fractional maximum f -de...

A strong edge coloring of a graph G is an edge coloring such that every two adjacent edges or two edges adjacent to a same edge receive two distinct colors; in other words, every path of length three has three distinct colors in G. The strong chromatic index of G, denoted by   S G  , is the smallest integer k such that G admits a strong edge coloring with k colors. This survey is an brief i...

In [4], Cariolaro et al. demonstrated how colorability problems can be approached numerically by the use of computer algebra systems and the Combinatorial Nullstellensatz. In particular, they verified a case of the List Coloring Conjecture by proving that the list-chromatic index of K6 is 5. In this short note, we show that using the coefficient formula of Schauz [16] is much more efficient tha...

The Classification Problem is the problem of deciding whether a simple graph has chromatic index equals to ∆ or ∆+1, where ∆ is the maximum degree of the graph. It is known that to decide if a graph has chromatic index equals to ∆ is NP-complete. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, s...