Graphs with the Circuit Cover Property

Alspach, Brian; Goddyn, Luis; Zhang, Cun-Quan

doi:10.2307/2154711

Cited by 5 publications

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References 2 publications

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“…Edwards, Robertson, Sanders, Sey- Figure 1 mour and Thomas [RST97, RST15, RST14, SST,ESST14] have announced a proof that every bridgeless cubic P-minor-free graph is edge 3-colourable, which is equivalent to Tutte's conjecture in the cubic case. Alspach, Goddyn and Zhang [AGZ94] showed that a graph has the circuit cover property if and only if it has no P-minor. It is recognised that determining the structure of P-minor-free graphs is a key open problem in graph minor theory (see [DLM16,Mah98] for example).…”

Section: Petersen Minorsmentioning

confidence: 99%

The extremal function for Petersen minors

Hendrey

Wood

2018

Journal of Combinatorial Theory, Series B

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We prove that every graph with n vertices and at least 5n − 8 edges contains the Petersen graph as a minor, and this bound is best possible. Moreover we characterise all Petersen-minor-free graphs with at least 5n − 11 edges. It follows that every graph containing no Petersen minor is 9-colourable and has vertex arboricity at most 5. These results are also best possible. IntroductionA graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by the following operations: vertex deletion, edge deletion and edge contraction. The theory of graph minors, initiated in the seminal work of Robertson and Seymour, is at the forefront of research in graph theory. A fundamental question at the intersection of graph minor theory and extremal graph theory asks, for a given graph H, what is the maximum number ex m (n, H) of edges in an n-vertex graph containing no H-minor? The function ex m (n, H) is called the extremal function for H-minors.The extremal function is known for several graphs, including the complete graphs K 4 and K 5 [Wag37, Dir64], K 6 and K 7 [Mad68], K 8 [Jør94] and K 9 [ST06], the bipartite graphs K 3,3 [Hal43] and K 2,t [CRS11], and the octahedron K 2,2,2 [Din13], and the complete graph on eight vertices minus an edge K − 8 [Son05]. Tight bounds on the extremal function are known for general complete graphs K t [dlV83, Kos82, Kos84, Tho84, Tho01], unbalanced complete bipartite graphs K s,t [KP08, KP10, KP12, KO05], disjoint unions of complete graphs [Tho08], disjoint unions of cycles [HW15, CLN + 15], general dense graphs [MT05] and general sparse graphs [RW16, HW16].

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