Triangle-free subgraphs with large fractional chromatic number

Mohar, Bojan; Wu, Hehui

doi:10.1016/j.endm.2015.06.089

Cited by 2 publications

(4 citation statements)

References 16 publications

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“…They are based on two concepts, that of a principal subset of vertices and that of a sparse subset. These two notions were used previously in our fractional versions of the Erdős‐Neumann‐Lara conjecture (see [19]) and the Erdős‐Hajnal conjecture from [12] for triangle‐free subgraphs (see [20]) that every graph with large chromatic number contains a triangle‐free subgraph whose chromatic number is still large.…”

Section: Fractional Weight and Principal Vertex‐setsmentioning

confidence: 99%

Fractional chromatic number of a random subgraph

Mohar

2020

Journal of Graph Theory

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It is well known [6] that a random subgraph of the complete graph K n has chromatic number Θ(n/ log n) w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any n-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least n, the fractional chromatic number of its random subgraph is at least n/(8 log 2 (4n)) with probability more than 1 − 1 2n . This gives the affirmative answer for a strengthening of Bukh's question for the fractional chromatic number.

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Section: Fractional Weight and Principal Vertex‐setsmentioning

confidence: 99%

Fractional chromatic number of a random subgraph

Mohar

2020

Journal of Graph Theory

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“…No capítulo 3 mostraremos que os grafos de Kneser respeitam a conjectura de Erdős e Hajnal [17], especificamente mostraremos que grafos de Kneser suficientemente grandes contém blow-ups de grafos de Kneser menores como subgrafos, e usando o Lema Local de Lovász mostraremos que blow-ups de grafos de Kneser contém subgrafos com número cromático e cintura grandes.…”

Section: 0unclassified

“…No Capítulo 3 vimos que a família dos grafos de Kneser respeitam a conjectura de Erdős e Hajnal [17]. Em particular mostramos por meio do Lema Local de Lovász que blow-ups suficientemente grandes de grafos de Kneser contém subgrafos com número cromático e cintura grandes, e também mostramos que blow-ups de grafos de Kneser são subgrafos de grafos de Kneser maiores.…”

Section: Considerações Finaisunclassified

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Uma conjectura de Erdos e Hajnal

Enju¹

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A result by Erdős shows that there exist graphs with arbitrarily large chromatic number and girth. Thus a sufficiently large clique contains a subgraph with large chromatic number and girth, but many graphs do not have a large clique, hence it is interesting to find a different condition that guarantees the existence of a subgraph with large chromatic number and girth. A conjecture by Erdős and Hajnal states that every graph with sufficiently large chromatic number contains a subgraph with large chromatic number and girth. The objective of this text is to study this conjecture.The text begins with a brief discussion of triangle-free constructions. In particular, we show a construction by Codenotti, Pudlák and Resta, based on projective planes.The main topic begins with a proof by Rödl, that every graph with sufficiently large chromatic number contains a triangle-free subgraph with large chromatic number. We follow with a proof that every graph with sufficiently large chromatic number contains a large odd cycle.We then show a result by Mohar and Wu, which shows that the Kneser graphs respect the Erdős-Hajnal conjecture. Another result by Gábor Tardos proves that shift graphs also respect the Erdős-Hajnal conjecture. Finally, we show some brief contributions about type graphs, showing some cases that follow the Erdős-Hajnal conjecture.

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Triangle-free subgraphs with large fractional chromatic number

Cited by 2 publications

References 16 publications

Fractional chromatic number of a random subgraph

Fractional chromatic number of a random subgraph

Uma conjectura de Erdos e Hajnal

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