Partial Degree Conditions and Cycle Coverings

2014 ◽  
Vol 78 (4) ◽  
pp. 295-304 ◽  
Author(s):  
Hong Wang
Keyword(s):  
Partial Degree ◽  
Degree Conditions

Generalized degree conditions for graphs with bounded independence number

1995 ◽  
Vol 19 (3) ◽  
pp. 397-409 ◽  
Author(s):  
Ralph Faudree ◽  
Ronald J. Gould ◽  
Linda Lesniak ◽  
Terri Lindquester
Keyword(s):  
Independence Number ◽  
Degree Conditions

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

scholarly journals Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions

10.37236/7353 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Jinko Kanno ◽  
Songling Shan
Keyword(s):  
Maximum Degree ◽  
Hamiltonian Cycle ◽  
Simple Graph ◽  
Chromatic Index ◽  
New Techniques ◽  
Critical Graph ◽  
Degree Conditions ◽  
Delta G ◽  
Proper Subgraph

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\chi'(G)=\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then  $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+13$, then $G$  has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and  maximum degree conditions. In particular, we show that  if $G$ is an  $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3/2$, but have no 2-factor. This implies that the $\Delta$-criticality in the result is needed. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

scholarly journals Maximum and Minimum Degree Conditions for Embedding Trees

2020 ◽  
Vol 34 (4) ◽  
pp. 2108-2123
Author(s):  
Guido Besomi ◽  
Matías Pavez-Signé ◽  
Maya Stein
Keyword(s):  
Minimum Degree ◽  
Degree Conditions ◽  
Maximum And Minimum Degree

A Note on Degree Conditions for Traceability in Locally Claw-Free Graphs

2011 ◽  
Vol 5 (1) ◽  
pp. 21-25 ◽  
Author(s):  
Roman Čada ◽  
Evelyne Flandrin ◽  
Haiyan Kang
Keyword(s):  
Degree Conditions

scholarly journals Degree conditions for forests in graphs

2005 ◽  
Vol 301 (2-3) ◽  
pp. 228-231 ◽  
Author(s):  
Ch. Sobhan Babu ◽  
Ajit A. Diwan
Keyword(s):  
Degree Conditions
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