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)

An edge coloring of a graph G with colors 1, 2, . . . , t is called an interval t-coloring if for each i ∈ {1, 2, . . . , t} there is at least one edge of G colored by i, the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1994 Asratian and Kamalian proved that if a connected graph G admits an interval t-coloring, then t ≤ (d + 1) (∆ − 1) + 1, and if G is also bipartite, then this upper bound can be improved to t ≤ d (∆ − 1) + 1, where ∆ is the maximum degree in G and d is the diameter of G. In this paper we show that these upper bounds can not be significantly improved.

For bipartite graphs the NP-completeness is proved for the problem of existence of maximum matching which removal leads to a graph with given lower(upper)bound for the cardinality of its maximum matching.Comment: 12 pages, 8 figures. Discrete Mathematics, to appea