Proof of a conjecture on induced subgraphs of Ramsey graphs

An n-vertex graph G is c-Ramsey if it contains neither a complete nor an empty induced subgraph of size greater than c log n. Erdős, Faudree and Sós conjectured that every c-Ramsey graph with n vertices contains Ω(n5/2) induced subgraphs, any two of which differ either in the number of vertices or in the number of edges, i.e., the number of distinct pairs (|V(H)|, |E(H)|), as H ranges over all induced subgraphs of G, is Ω(n5/2). We prove an Ω(n2.3693) lower bound.

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Induced subgraphs of Ramsey graphs with many distinct degrees

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2006.09.006 ◽

2007 ◽

Vol 97 (4) ◽

pp. 612-619 ◽

Cited By ~ 6

Author(s):

Boris Bukh ◽

Benny Sudakov

Keyword(s):

Induced Subgraphs ◽

Ramsey Graphs

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Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2021.02.007 ◽

2021 ◽

Vol 295 ◽

pp. 57-69

Author(s):

A. Atminas ◽

R. Brignall ◽

V. Lozin ◽

J. Stacho

Keyword(s):

Induced Subgraphs ◽

Forbidden Induced Subgraphs

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Excluding induced subgraphs: Critical graphs

Random Structures and Algorithms ◽

10.1002/rsa.20353 ◽

2010 ◽

Vol 38 (1-2) ◽

pp. 100-120 ◽

Cited By ~ 10

Author(s):

József Balogh ◽

Jane Butterfield

Keyword(s):

Induced Subgraphs ◽

Critical Graphs

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Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830914500438 ◽

2014 ◽

Vol 06 (03) ◽

pp. 1450043

Author(s):

Bo Ning ◽

Shenggui Zhang ◽

Bing Chen

Keyword(s):

Hamilton Cycles ◽

Induced Subgraphs ◽

Degree Sum ◽

Degree Condition

Let claw be the graph K1,3. A graph G on n ≥ 3 vertices is called o-heavy if each induced claw of G has a pair of end-vertices with degree sum at least n, and called 1-heavy if at least one end-vertex of each induced claw of G has degree at least n/2. In this note, we show that every 2-connected o-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman and Hedetniemi, Gould and Jacobson, and Shi on the existence of Hamilton cycles in graphs.

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