No 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)

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3, 4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide sufficient conditions for the existence of such a subgraph.

No abstract

SUMMARYStarting from the simple class-teacher model of timetabling (where timetables correspond to edge colorings of a bipartite multigraph), we consider an extension deÿned as follows: we assume that the set of classes is partitioned into groups. In addition to the teachers giving lectures to individual classes, we have a collection of teachers who give all their lectures to groups of classes. We show that when there is one such teacher giving lectures to three groups of classes, the problem is NP-complete. We also examine the case where there are at most two groups of classes and we give a polynomial procedure based on network ows to ÿnd a timetable using at most t periods.

A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree ∆(G) ≥ 4 admits a cyclic intervalWe also prove that every Eulerian bipartite graph G with maximum degree at most 8 has a cyclic interval coloring. Some results are obtained for (a, b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r −2, 2r)-biregular (r ≥ 2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

For an integer i, a graph is called an Li‐graph if, for each triple of vertices u, v, w with d(u, v) = 2 and w (element of) N(u) (intersection) N(v), d(u) + d(v) ≥ | N(u) (union) N(v) (union) N(w)| —i. Asratian and Khachatrian proved that connected Lo‐graphs of order at least 3 are hamiltonian, thus improving Ore's Theorem. All K1,3‐free graphs are L1‐graphs, whence recognizing hamiltonian L1‐graphs is an NP‐complete problem. The following results about L1‐graphs, unifying known results of Ore‐type and known results on K1,3‐free graphs, are obtained. Set K = lcub;G|Kp,p+1 (contained within) G (contained within) Kp V Kp+1 for some ρ ≥ } (v denotes join). If G is a 2‐connected L1‐graph, then G is 1‐tough unless G (element of) K. Furthermore, if G is as connected L1‐graph of order at least 3 such that |N(u) (intersection) N(v)| ≥ 2 for every pair of vertices u, v with d(u, v) = 2, then G is hamiltonian unless G ϵ K, and every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path. This result implies that of Asratian and Khachatrian. Finally, if G is a connected L1‐graph of even order, then G has a perfect matching. © 1996 John Wiley & Sons, Inc.

No abstract