Small 1-defective Ramsey numbers in perfect graphs

Ekim, Tınaz; Gimbel, John; Şeker, Oylum

doi:10.1016/j.disopt.2019.06.001

Cited by 5 publications

(12 citation statements)

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“…Our first result is related to R PG 1 (4, 8); its value has been shown to be 15 in [9], however only one extremal graph (the Heawood graph depicted in Figure 1) has been reported. Here, we show that there are exactly three extremal graphs.…”

Section: Defective Ramsey Numbers In Perfect Graphsmentioning

confidence: 72%

“…So assume that G has no 1-dense 4-set and we will prove that G has a 1-sparse 10-set. As already stated, we know G has no induced subgraph on 18 vertices that is isomorphic to G 4 , the unique graph in T PG 18 (1,4,9). Thus, every induced subgraph of G on 18 vertices has a 1-sparse 9-set.…”

Section: Defective Ramsey Numbers In Perfect Graphsmentioning

confidence: 84%

“…Besides efficient graph generation algorithms, a recent trend is to consider defective Ramsey numbers in graph classes. This has been initiated in [9] where some 1-defective Ramsey numbers in perfect graphs have been computed. Further graph classes have been examined in [7] from the perspective of defective Ramsey numbers; namely, formulas for all or most cases have been established for defective Ramsey numbers in forests, cacti, bipartite graphs, split graphs and cographs, and the missing cases have been formulated as conjectures.…”

Section: Introductionmentioning

confidence: 99%

“…All of our codes and the extremal graphs we obtain are available at https://github.com/yunusdemirci/DefectiveRamsey [18]. By combining our efficient graph generation algorithm with classical proof techniques, we extend the results in [7] and [9] by establishing new defective Ramsey numbers in perfect graphs and bipartite graphs, and investigate chordal graphs which have not been addressed from this perspective to the best of our knowledge. In what follows, let PG, BIP and CH denote the classes of perfect, bipartite and chordal graphs, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3, we follow up the results in [9] by establishing new defective Ramsey numbers in perfect graphs. First, we give all extremal graphs for R PG 1 (4, 8) (whose value has been already shown in [9] without providing all extremal graphs), and then establish that R PG 1 (4, 9) = 19 and R PG 1 (4, 10) = 22. We also compute several k-defective Ramsey numbers in perfect graphs for k ∈ {2, 3, 4} as reported in Tables 2, 3 and 4 respectively.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs

Demirci¹,

Ekim²,

Yıldız³

2021

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In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to k-dense and k-sparse sets, both commonly called k-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs. Since direct proof techniques are often insufficient for obtaining new values of defective Ramsey numbers, we provide a generic algorithm to compute defective Ramsey numbers in a given target graph class. We combine direct proof techniques with our efficient graph generation algorithm to compute several new defective Ramsey numbers in perfect graphs, bipartite graphs and chordal graphs. We also initiate the study of a related parameter, denoted by c G k (m), which is the maximum order n such that the vertex set of any graph of order at n in a class G can be partitioned into at most m subsets each of which is k-defective. We obtain several values for c G k (m) in perfect graphs and cographs.

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Section: Defective Ramsey Numbers In Perfect Graphsmentioning

confidence: 72%

Section: Defective Ramsey Numbers In Perfect Graphsmentioning

confidence: 84%