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.

It was observed by Dulmage and Mendelsohn in their work on matrix reducibility that there is a one-to-one correspondence between bigraphs and digraphs determined by the utilization of the adjacency matrix. In this semiexpository paper we explore the interaction between this correspondence and a theory of matrix decomposability that is developed in several different articles. These results include: (a) a characterization of those bipartite graphs that can be labeled so that the resulting digraph is symmetric; (b) a criterion for the bigraph of a symmetric digraph to be connected; (c) a necessary and sufficient condition for a square binary matrix to be fully indecomposable in terms of its associated bigraph, and (d) matrix criteria for a digraph to be strongly, unilaterally, or weakly connected. We close with an unsolved extermal problem on the number of components of the bigraph of various orientations of a given graph. This leads to new amusing characterizations of trees and bigraphs.Dedicated to the graph-theoretic partnership of Lloyd Dulmage and Nathan Mendelsohn.

We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference I V(G)I -I V(H)I over graphs G having center isomorphic to H it is demonstrated that A(H)54 for all H, and for 0 5 i 5 4 we characterize the class of trees T withA(T) = i. For n 2 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n , with chromatic number x(G) = n 4-x(H), and whose center is isomorphic to H. Finally, if I V(H) I 2 9 and k 2 I V(H) I + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.

The biparticity P(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, P ( G ) = {10g2x(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.Dedicated to the memory of Paul Turhn BlPARTlClTYThis graphical invariant was defined in [3] along with several other concepts relating to coverings and packings of graphs. The biparticity P(G) is the minimum number of spanning bipartite subgraphs which cover the lines of G. We will determine the biparticity of any graph G in terms of its chromatic number.Our notation and terminology will follow the book [2]. As usual, write {x} for the least integer not less than the real number x. Let E ( G ) be the set of lines of G, V(G) the set of points, and p(@) the number of points. THEOREM. For any graph G, P ( G )Proof. For the upper bound we will proceed by induction to show that if x(G) = 2", then P(G) I n. This is obviously true if n = 1. Assume it is true for n = k, and that x(G) = 2'". Let G, be the subgraph generated by 2' of the colors, and let Gz be the subgraph generated by the other 2' colors. Then x ( Gi) = 2k. By induction p ( G) 5 1 f max P ( G i ) 5 1 + k as required.

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.