ABSTRACT:We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edgecoloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G ϭ K t , we have c 1 mt 2 /ln t Յ f(m, K t ) Յ c 2 mt 2 , and cЈ 1 mt 3 /ln t Յ g(m, K t ) Յ cЈ 2 mt 3 /ln t, where c 1 , c 2 , cЈ 1 , cЈ 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) ϭ n for all graphs G with n vertices and maximum degree at most d.
We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.
We consider the following graph labeling problem, introduced by Leung et al. (J. Y‐T. Leung, O. Vornberger, and J. D. Witthoff, On some variants of the bandwidth minimization problem. SIAM J. Comput. 13 (1984) 650–667). Let G be a graph of order n, and f a bijection from V(G) to the integers 1 through n. Let |f|, and define s(G), the separation number of G, to be the maximum of |f| among all such bijections f. We first derive some basic relations between s(G) and other graph parameters. Using a general strategy for analyzing separation number in bipartite graphs, we obtain exact values for certain classes of forests and asymptotically optimal lower bounds for grids and hypercubes.
In this paper we investigate simulations of hypercube networks by certain Cayley graphs on the symmetric group. Let Q(k) be the familiar k-dimensional hypercube, and let S(n) be the star
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