Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Abstract

A strong edge coloring of a graph G is a proper edge coloring of G such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 4 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by 4.

Graphs Defined on Rings: A Review

“…This motivated researchers to study different colourings of the graph and to investigate the related parameters and properties. The total colouring and the strong edge colouring of unitary Cayley graphs were studied in [103][104][105]. A total colouring of a graph G is a proper colouring on both the edges and vertices, such that no two adjacent entities (both vertices and edges) are assigned the same colour, and the total chromatic number is the minimum number of colours required in the total colouring of the graph (see [105]).…”

“…A strong edge colouring of a graph G is a proper edge colouring of G such that every colour class induces a matching, and the minimum number of colours required is the strong chromatic index. In [104], the strong chromatic index of all unitary Cayley graphs was determined, and the colouring technique revealed the underlying product structure from which the unitary Cayley graphs emerge.…”