منابع مشابه
Squares of Low Clique Number
A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. This problem is only known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very l...
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Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem o...
متن کاملClique number of random Cayley graph
Let G be a finite abelian group of order n. For any subset B of G with B = −B, the Cayley graph GB is a graph on vertex set G in which ij is an edge if and only if i − j ∈ B. It was shown by Ben Green [3] that when G is a vector space over a finite field Z/pZ, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than c logn log logn, where c > ...
متن کاملClique decompositions of multipartite graphs and completion of Latin squares
Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the...
متن کامل7.1 Clique Number of Random Graphs
(Throughout this section, log denotes base-2 logarithm.) We now sketch a proof of this theorem. Proof: Let Xk be the number of k-cliques in a graph G drawn from Gn, 2 . By linearity of expectation we have g(k) := E[Xk] = ( n k ) 2−( k 2) and let us define k0(n) to be the largest value of k such that g(k) ≥ 1. An easy calculation (Exercise!) shows that k0(n) ∼ 2 log n. (To see this is plausible,...
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ژورنال
عنوان ژورنال: Electronic Notes in Discrete Mathematics
سال: 2016
ISSN: 1571-0653
DOI: 10.1016/j.endm.2016.10.048