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Positive Integer K (positive + integer_k)
Selected AbstractsMulti-coloring the Mycielskian of graphsJOURNAL OF GRAPH THEORY, Issue 4 2010Wensong Lin Abstract A k -fold coloring of a graph is a function that assigns to each vertex a set of k colors, so that the color sets assigned to adjacent vertices are disjoint. The kth chromatic number of a graph G, denoted by ,k(G), is the minimum total number of colors used in a k -fold coloring of G. Let µ(G) denote the Mycielskian of G. For any positive integer k, it holds that ,k(G) + 1,,k(µ(G)),,k(G) + k (W. Lin, Disc. Math., 308 (2008), 3565,3573). Although both bounds are attainable, it was proved in (Z. Pan, X. Zhu, Multiple coloring of cone graphs, manuscript, 2006) that if k,2 and ,k(G),3k,2, then the upper bound can be reduced by 1, i.e., ,k(µ(G)),,k(G) + k,1. We conjecture that for any n,3k,1, there is a graph G with ,k(G)=n and ,k(µ(G))=n+ k. This is equivalent to conjecturing that the equality ,k(µ(K(n, k)))=n+k holds for Kneser graphs K(n, k) with n,3k,1. We confirm this conjecture for k=2, 3, or when n is a multiple of k or n,3k2/ln k. Moreover, we determine the values of ,k(µ(C2q+1)) for 1,k,q. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 311,323, 2010 [source] Unavoidable parallel minors of 4-connected graphsJOURNAL OF GRAPH THEORY, Issue 4 2009Carolyn Chun A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k -partition triple fan with a complete graph on the vertices of degree k, the k -spoke double wheel, the k -spoke double wheel with axle, the (2k+1)-rung Möbius zigzag ladder, the (2k)-rung zigzag ladder, or Kk. We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 313-326, 2009 [source] More efficient queries in PCPs for NP and improved approximation hardness of maximum CSPRANDOM STRUCTURES AND ALGORITHMS, Issue 4 2008Lars Engebretsen Abstract Samorodnitsky and Trevisan [STOC 2000, pp. 191,199] proved that there exists, for every positive integer k, a PCP for NP with O(log n) randomness, query complexity 2k + k2, free bit complexity 2k, completeness 1 - ,, and soundness 2 + ,. In this article, we devise a new "outer verifier," based on the layered label cover problem recently introduced by Dinur et al. [STOC 2003, pp. 595,601], and combine it with a new "inner verifier" that uses the query bits more efficiently than earlier verifiers. Our resulting theorem is that there exists, for every integer f , 2, every positive integer t , f(f - 1)/2, and every constant , > 0, a PCP for NP with O(log n) randomness, query complexity f + t, free bit complexity f, completeness 1 - ,, and soundness 2 - t + ,. As a corollary, there exists, for every integer q , 3 and every constant , > 0, a q -query PCP for NP with amortized query complexity 1 + + ,. This improves upon the result of Samorodnitsky and Trevisan with respect to query efficiency, i.e., the relation between soundness and the number of queries. Although the improvement may seem moderate,the construction of Samorodnitsky and Trevisan has amortized query complexity 1 + 2/,we also show in this article that combining our outer verifier with any natural candidate for a corresponding inner verifier gives a PCP that is less query efficient than the one we obtain.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 [source] On the complexity of the circular chromatic numberJOURNAL OF GRAPH THEORY, Issue 3 2004H. Hatami Abstract Circular chromatic number, ,c is a natural generalization of chromatic number. It is known that it is NP -hard to determine whether or not an arbitrary graph G satisfies ,(G)=,c(G). In this paper we prove that this problem is NP -hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k,,,2 and n,,,3, for a given graph G with ,(G),=,n, it is NP -complete to verify if . © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 226,230, 2004 [source] | ||||||||||