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An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label seq...
The vertex Folkman number F(r, n, m), n < m, is the smallest integer t such that there exists a Km-free graph of order t with the property that every r-coloring of its vertices...
The linear-width of a graph G is de ned to be the smallest integer k such that the edges of G can be arranged in a linear ordering e1;:::;er in such a way that for every i = 1;:::...
For the unbiased Maker-Breaker game, played on the hypergraph H, let τM (H) be the smallest integer t such that Maker can win the game within t moves (if the game is a Breaker’...
Dan Hefetz, Michael Krivelevich, Milos Stojakovic,...
: For two given graphs G1 and G2, the Ramsey number R(G1, G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G...
T. C. Edwin Cheng, Yaojun Chen, Yunqing Zhang, C. ...
A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest...
Vikraman Arvind, Christine T. Cheng, Nikhil R. Dev...
1 The Ramsey number R(G1, G2) is the smallest integer p such that for any graph G on p vertices2 either G contains G1 or G contains G2, where G denotes the complement of G. In this...
In this note we report on our recent work, still in progress, regarding Folkman numbers. Let f(2, 3, 4) denote the smallest integer n such that there exists a K4
Given any integer d ≥ 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k + 1...
Given d (0, ) let kd be the smallest integer k such that d < 2k log k. We prove that the chromatic number of a random graph G(n, d/n) is either kd or kd + 1 almost surely.