Hardness results and spectral techniques for combinatorial problems on circulant graphs

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r f (a 1 , a 2 , . . . , a k ) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k = 2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. ᭧

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“…To illustrate our construction, let (a 1 , a 2 , a 3 ) = (5,6,8). Then, we derive (n 1 , n 2 , n 3 ) = (4,23,183 …”

Section: S Ti For J = Imentioning

confidence: 99%

“…However, in [5], it is proven that determining the independence number of an arbitrary circulant graph is NP-hard. Moreover, it is shown [5] that it is NP-hard even to get a good approximation for (G). A formula for (C n,S ) is known only for a handful of generating sets.…”

Section: Introductionmentioning

confidence: 99%

Proof of a conjecture on fractional Ramsey numbers

Brown

Hoshino

2009

Journal of Graph Theory

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“…Our results have obvious algorithmic consequences, since polynomial-time algorithms for graph isomorphism were constructed for many classes of graphs, including trees (Aho, Hopcroft, and Ullman), graphs with bounded vertex degrees (Luks [36], Furst et al [23], and Galil et al [24]), planar graphs (Hopcroft and Tarjan [27,28], Gazit [25], and Ja'Ja' and Kosaraju [29]), interval graphs (Lueker and Booth [35] and Klein [30]), graphs with bounded eigenvalue multiplicities (Babai [4] and Fürer [22]), partial k-trees (Bodlaender [9,10]), graphs with bounded average genus (Chen [15]), convex bipartite graphs (Chen [16]), circulant graphs (Codenotti et al [17]), graphs of bounded distance width (Yamazakiet al [45]), and others, see Hoffmann [26], Köbler et al [31], Babai [5], Campbell and Radford [13], Chen [16], Fukuda and Nakamori [21], Ponomarenko [40,41], Rasin [42], etc.…”

Section: Corollary 6 the Bipartite Fixed Neighborhood Realization Prmentioning

confidence: 99%

Neighborhood hypergraphs of bipartite graphs

Boros

Gurvich

Zverovich

2008

Journal of Graph Theory

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Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP-hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP-hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial-time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs.

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“…In the last years, it had been often investigated if a graph problem becomes easier when it is restricted to the circulant graphs. For example, the Maximum Clique problem, and the Minimum Graph Coloring problem remain NP-hard, and not approximable within a constant factor, when the general instance is forced to be a circulant undirected graphs, as shown by Codenotti, et al (1998). On the other hand, Muzychuk (2004) has proved that the Graph Isomorphism problem restricted to circulant undirected graphs is in P, while the general case is, probably, harder.…”

Section: Introductionmentioning

confidence: 99%