Investigation on Interval Edge-Colorings of Graphs

Abstract

An edge-coloring of a simple graph G with colors !, 2, ... , t is called an interval t-coloring [3] if at least one edge of G is colored by color i, i = 1, ... , t and the edges incident with each vertex x are colored by dG(x) consecutive colors, where dG(x) is the degree of the vertex x. In this paper we investigate some properties of interval colorings and their variations. It is proved, in particular, that if a simple graph G = (V, E) without triangles has an interval t-coloring, then t,;; I VI-1. 1;, 1994 Academic Press, Jnc. 1. INTRODUCTION All graphs considered in this paper are undirected and have no loops or multiple edges. V(G) and E(G) denote the sets of vertices and edges of a graph G, respectively. The degree of a vertex x in G is denoted by dG(x), and the diameter of G is denoted by d(G). A t-coloring of a graph G is a function f: £(G)-+ { 1, ... , t}, such that f(e) i= f (e') for any pair of adjacent edges e and e'. If eEE(G) and f(e)=i then e is said to be colored by color i. The chromatic index x'(G) is the least value of t for which a t-coloring of G exists. A well-known theorem of V. G. Vizing [ 17] states that Ll(G)<x'(G)<Ll(G)+l where Ll(G) denotes the maximum degree of G. If x'(G) = Ll(G) then G is said to be Class 1 and otherwise G is Class 2. The problem of deciding whether or not a regular graph is Class 1 was shown by I. Holyer [10] to be %&'-complete (see also [15]). Let R i:;;. V(G). An interval (R, t)-coloring of G is a t-coloring of G such that at least one edge of G is colored by color i, i = 1, ... , t, and the edges incident with each x ER are colored by dG(x) consecutive colors. An interval t-coloring of G is an interval (V(G), t)-coloring of G.

Interval edge-colorings of complete graphs

“…In [1,2], they proved that if G is interval colorable, then χ (G) = ∆(G). For regular graphs the converse is also true.…”

“…In [6] Kamalian proved the following upper bound on W (G): Improved upper bounds on W (G) are known for several classes of graphs, including triangle-free graphs [1,2], planar graphs [3] and r-regular graphs with at least 2r +2 vertices [7]. The exact value of the parameter W (G) is known for even cycles, trees [5], complete bipartite graphs [5], Möbius ladders [10] and n-dimensional cubes [11,12].…”

Interference-Free Wakeup Scheduling with Consecutive Constraints in Wireless Sensor Networks

“…Interval edge coloring, introduced by Asratian and Kamalian [20] (available in English as [21]), is a special edge coloring in which the colors of edges incident to the same vertex must be contiguous, that is, the colors must be composed of an integer interval. Not every graph has an interval edge coloring, since a graph G with an interval edge coloring belongs to Class 1 graphs where the chromatic number of edge coloring is equal to the maximum degree Δ [21]. Sevastjanov [22] proves that the problem of determining the existence of an interval edge coloring is NP-complete, even for bipartite graphs, and Kubale [23] proves that the interval edge coloring problem with forbidden colors is also NP-complete.…”

Interval edge-colorings of composition of graphs

“…In [ 1], they proved that if G ∈ N, then χ ′ (G) = ∆(G). Asratian and Kamalian also proved [ 1,2] that if a trianglefree graph G admits an interval t-coloring, then t ≤ |V (G)| − 1. In [ 16,17], Kamalian investigated interval colorings of complete bipartite graphs and trees.…”

“…In [ 31], Sevast'janov proved that it is an NP -complete problem to decide whether a bipartite graph has an interval coloring or not. In papers [ 1,2,6,7,9,16,17,20,24,26,27,28,31], the problems of existence, construction and estimating the numerical parameters of interval colorings of graphs were investigated. Surveys on this topic can be found in some books [ 3,15,20].…”