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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… (More)

Given a graph G and a subset W ⊆ V (G), a Steiner W-tree is a tree of minimum order that contains all of W. Let S(W) denote the set of all vertices in G that lie on some Steiner W-tree; we call S(W) the Steiner interval of W. If S(W) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given… (More)

—Given positive integers and , let 2 () denote the maximum size of a binary code of length and minimum distance . The well-known Gilbert–Varshamov bound asserts that 2 () 2 (1), where () = =0 is the volume of a Hamming sphere of radius. We show that, in fact, there exists a positive constant such that 2 () 2 (1) log 2 (1) whenever 0 499. The result follows… (More)

Given two graphs G and H, let f (G,H) denote the minimum integer n such that in every coloring of the edges of K n , there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f (G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we… (More)

In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation ⊕ for making a convexity space out… (More)

conjectured that every graph G with girth at least 2t+1 and minimum degree at least kÂt contains every tree T with k edges whose maximum degree does not exceed the minimum degree of G. The conjecture has been proved for t3. In this paper, we prove Dobson's conjecture. 2001 Elsevier Science

The chromatic sum Σ(G) of a graph G is the smallest sum of colors among all proper colorings with natural numbers. The strength s(G) of G is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree T k with strength k that has maximum degree only 2k − 2. The result is best possible.

A mixed hypergraph is a triple H = (X, C, D), where X is the vertex set, and each of C, D is a list of subsets of X. A strict k-coloring of H is a surjection c : X → {1,. .. , k} such that each member of C has two vertices assigned a common value and each member of D has two vertices assigned distinct values. The feasible set of H is {k : H has a strict… (More)