Planar graphs without triangles adjacent to cycles of length from 3 to 9 are 3-colorable.
Borodin, O.V., Glebov, A.N., Jensen, Tommy R., Raspaud, Andre (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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Borodin, O.V., Glebov, A.N., Jensen, Tommy R., Raspaud, Andre (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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Gould, Ronald, Łuczak, Tomasz, Schmitt, John (2006)
The Electronic Journal of Combinatorics [electronic only]
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Tomoki Yamashita (2007)
Discussiones Mathematicae Graph Theory
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A cycle C is a vertex-dominating cycle if every vertex is adjacent to some vertex of C. Bondy and Fan [4] showed that if G is a 2-connected graph with δ(G) ≥ 1/3(|V(G)| - 4), then G has a vertex-dominating cycle. In this paper, we prove that if G is a 2-connected bipartite graph with partite sets V₁ and V₂ such that δ(G) ≥ 1/3(max{|V₁|,|V₂|} + 1), then G has a vertex-dominating cycle.
Eckhard Steffen (2001)
Mathematica Slovaca
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Peled, Uri N., Wu, Julin (1996)
The Electronic Journal of Combinatorics [electronic only]
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Dragoš M. Cvetković, Slobodan K. Simić (1978)
Publications de l'Institut Mathématique
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Binlong Li, Liming Xiong, Jun Yin (2016)
Discussiones Mathematicae Graph Theory
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In this paper, we consider the least integer d such that every longest cycle of a k-connected graph of order n (and of independent number α) contains all vertices of degree at least d.
Vasil Jacoš (1975)
Matematický časopis
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