NDLI: Snarks, Hypohamiltonian Graphs and Non-Supereulerian Graphs

Content Provider	Springer Nature Link
Author	Chen, Zhi Hong
Copyright Year	2016
Abstract	A graph G is hypohamiltonian if it is not Hamiltonian but for each $$v\in V(G)$$ , the graph $$G-v$$ is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset $$R\subseteq V(G)$$ , there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173–181, 1990] and in [J. Combin. Theory, Ser B 66:123–139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291–304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.
Starting Page	2267
Ending Page	2273
Page Count	7
File Format	PDF
ISSN	09110119
Journal	Graphs and Combinatorics
Volume Number	32
Issue Number	6
e-ISSN	14355914
Language	English
Publisher	Springer Japan
Publisher Date	2016-05-20
Publisher Place	Tokyo
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hypohamiltonian graphs Snarks Supereulerian graphs Collapsible graphs Combinatorics Engineering Design
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Theoretical Computer Science

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