Displaying similar documents to “A Note on Barnette’s Conjecture”

On the Erdős-Gyárfás Conjecture in Claw-Free Graphs

Pouria Salehi Nowbandegani, Hossein Esfandiari, Mohammad Hassan Shirdareh Haghighi, Khodakhast Bibak (2014)

Discussiones Mathematicae Graph Theory

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The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs

On Vertices Enforcing a Hamiltonian Cycle

Igor Fabrici, Erhard Hexel, Stanislav Jendrol’ (2013)

Discussiones Mathematicae Graph Theory

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A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.

A Survey of the Path Partition Conjecture

Marietjie Frick (2013)

Discussiones Mathematicae Graph Theory

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The Path Partition Conjecture (PPC) states that if G is any graph and (λ1, λ2) any pair of positive integers such that G has no path with more than λ1 + λ2 vertices, then there exists a partition (V1, V2) of the vertex set of G such that Vi has no path with more than λi vertices, i = 1, 2. We present a brief history of the PPC, discuss its relation to other conjectures and survey results on the PPC that have appeared in the literature since its first formulation in 1981.