We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an algorithm that finds a coloring with discrepancy O((t log n log s)) where s is the maximum cardinality of a set. This improves upon the previous constructive bound of O(t logn) based on algorithmic variants of the partial coloring method, and for small s (e....

Abstract For all positive integers s and t, Brown et. al [1] defined f(s, t) to be the smallest positive integer N such that every 2-coloring of [1, N ] has a monochromatic homothetic copy of {1, 1+ s, 1+ s+ t}. They proved that f(s, t) ≤ 4(s + t) + 1 for all s, t and that the equality holds in the case where both s/g 6≡ 0 (mod 4) and t/g 6≡ 0 (mod 4) with g = gcd(s, t) and in many other cases....

In this paper we prove that if a weighted symmetric digraph (~ G, c) has a mapping T : E( ~ G) → {0, 1} with T (xy) + T (yx) = 1 for all arcs xy in ~ G such that for each dicycle C satisfying 0 < |C|c(mod r) < maxxy∈E(~ G) c(xy) + c(yx) we have |C|c/|C|T ≤ r, then (~ G, c) has a circular r-coloring. Our result generalizes the work of Zhu (J. Comb. Theory, Ser. B, 86 (2002) 109-113) concerning t...

The Brualdi-Shen Conjecture on Eulerian Bipartite Tournaments states that any such graph can be decomposed into oriented 4-cycles. In this article we prove the balanced case of the mentioned conjecture. We show that for any 2n×2n bipartite graph G = (U ∪V,E) in which each vertex has n-neighbors with biadjacency matrixM (or its transpose), there is a particular proper edge coloring of a column p...

Fox–Grishpun–Pach showed that every 3-coloring of the complete graph on n vertices without a rainbow triangle contains a set of order Ω ( n log n ) which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. We show that every such colori...

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...

An r-component connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been well-studied for r = 1, in the case of trees, under the rubric of convex coloring , used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are disc...

Let G be an r-uniform hypergraph. The multicolor Ramsey number rk(G) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K n yields a monochromatic copy of G. Improving slightly upon results from [1], we prove that tk + 1 ≤ rk(K2,t+1) ≤ tk + k + 2, where the lower bound holds when t and k are both powers of a prime p. When t = 1, we improve the lower bo...

Beck and Fiala conjectured in 1981 that for any set system S of maximum degree t on a nite ground set X, a coloring : X ! f?1; +1g exists such that j(S)j = O(p t) holds for all S 2 S, where (S) = P x2S (X). We prove a weaker statement, namely that for any xed p 1, a coloring exists such that the pth degree average of j(S)j over S 2 S is O(p t). The result also holds if each set is assigned a no...

An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if G has an interval t-coloring for some positive integer t. Let N be the set of all interval colorable graphs. For a graph G ∈ N, the least and the greatest va...