On <img src=image/13430241_01.gif>-coloring and <img src=image/13430241_02.gif>-coloring ofWindmill Graph

Abstract

The windmill graph W (r, m); m ≥ 3, r ≥ 2 is the graph formed by joining a common vertex to every vertex of m copies of the complete graph K r . T -coloring of a graph is a map h defined on the set of vertices in such a way that for any edge (w 1 , w 2 ), | h(w 1 ) − h(w 2 ) | does not belong to a finite set T of non-negative integers. Strong T-Coloring (ST -coloring) is a particular case of T -coloring and is defined as the map:for any two distinct edges (w 1 , w 2 ), (w 3 , w 4 ) ∈ E(G). Application of T and ST -coloring of graph naturally arises in the modeling of different scientific problems. Frequency assignment problem (FAP) is one of the well known problems in the field of telecommunication, which can be modeled using the concept of T and ST -coloring of graphs. In this paper, we will consider two special types of T -sets. The first one is λ-initial set, introduced by Cozzens and Roberts, which is of the form {0, 1, 2, . . . , λ} ∪ S where S is any arbitrary set that doesn't contain any multiple of (λ + 1). The second one is λ-multiple of q set, introduced by Raychaudhuri, which is of the form {0, q, 2q, . . . , λq} ∪ S, where S is a subset of the set {q + 1, q + 2, q + 3, . . . , λq}. We will discuss some parameters related to these two types of colorings viz. T -chromatic number, T -span, T -edge span on the basis of the two T -sets. We will also deduce some generalized results of ST -coloring of any graph based on any T -set, and with the help of these results we will obtain ST -chromatic number and bounds for the ST -span and ST -edge span of windmill graphs.