A Theorem of Finite Sets

Katona, Gyula O. H.

doi:10.1007/978-0-8176-4842-8_27

Cited by 227 publications

(212 citation statements)

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“…The answer to this problem is a corollary to the Kruskal-Katona theorem [14,13] involving minimizing the shadow of a set system. This section provides an introduction to our techniques that will be employed in the case of the Widom-Rowlinson model.…”

Section: Estimating the Number Of Independent Setsmentioning

confidence: 99%

“…This is the content of Section 3. While this result is a corollary of the Kruskal-Katona theorem [14,13], we believe that seeing this example first will help in understanding the main proof. The remainder of the paper is devoted to a proof that the maximum value of wr(G) among graphs on n vertices having e edges is achieved by some graph coming from one of five families.…”

Section: Introductionmentioning

confidence: 99%

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Extremal graphs for homomorphisms

Cutler

Radcliffe

2011

J. Graph Theory

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Abstract. The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K2 with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P • 2 , the completely looped path of length 2 (known as the Widom-Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these "potentially extremal" threshold graphs is in fact extremal for some number of edges.

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Section: Estimating the Number Of Independent Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extremal graphs for homomorphisms

Cutler

Radcliffe

2011

J. Graph Theory

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“…This question was answered independently by Kruskal [11] and Katona [10] in 1960s. For a positive integer k, they enlisted all k-element subsets of integers in the following order, called the squashed order: A < B if max(A \ B) < max(B \ A).…”

Section: Kruskal-katona Theoremmentioning

confidence: 96%

$$f$$ -Vectors Implying Vertex Decomposability

Lasoń

2012

Discrete Comput Geom

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“…. Recall that i t (G) is the number of independent sets of size t in G. The following result is a consequence of the Kruskal-Katona theorem [41,40]. Theorem 9.7.…”

Section: Petersen Graphmentioning

confidence: 99%