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Article overview
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Solution to a problem on hamiltonicity of graphs under Ore- and Fan-type heavy subgraph conditions | | Bo Ning
; Shenggui Zhang
; Binlong Li
; | | Date: |
11 Sep 2014 | | Abstract: | A graph $G$ is called emph{claw-o-heavy} if every induced claw ($K_{1,3}$)
of $G$ has two end-vertices with degree sum at least $|V(G)|$ in $G$. For a
given graph $R$, $G$ is called emph{$R$-f-heavy} if for every induced subgraph
$H$ of $G$ isomorphic to $R$ and every pair of vertices $u,vin V(H)$ with
$d_H(u,v)=2$, there holds $max{d(u),d(v)}geq |V(G)|/2$. In this paper, we
prove that every 2-connected claw-emph{o}-heavy and $Z_3$-emph{f}-heavy graph
is hamiltonian (with two exceptional graphs), where $Z_3$ is the graph obtained
from identifying one end-vertex of $P_4$ (a path with 4 vertices) with one
vertex of a triangle. This result gives a positive answer to a problem proposed
in [B. Ning, S. Zhang, Ore- and Fan-type heavy subgraphs for Hamiltonicity of
2-connected graphs, Discrete Math. 313 (2013) 1715--1725], and also implies two
previous theorems of Faudree et al. and Chen et al., respectively. | | Source: | arXiv, 1409.3325 | | Services: | Forum | Review | PDF | Favorites | |
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