scholarly journals Hamilton cycles in almost distance-hereditary graphs

2016 ◽  
Vol 14 (1) ◽  
pp. 19-28
Author(s):  
Bing Chen ◽  
Bo Ning
Keyword(s):  
Hamiltonian Cycle ◽  
Previous Theorem ◽  
Hamilton Cycles ◽  
Induced Subgraph ◽  
Degree Sum ◽  
Degree Condition

AbstractLet G be a graph on n ≥ 3 vertices. A graph G is almost distance-hereditary if each connected induced subgraph H of G has the property dH(x, y) ≤ dG(x, y) + 1 for any pair of vertices x, y ∈ V(H). Adopting the terminology introduced by Broersma et al. and Čada, a graph G is called 1-heavy if at least one of the end vertices of each induced subgraph of G isomorphic to K1,3 (a claw) has degree at least n/2, and is called claw-heavy if each claw of G has a pair of end vertices with degree sum at least n. In this paper we prove the following two theorems: (1) Every 2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian. (2) Every 3-connected, 1-heavy and almost distance-hereditary graph is Hamiltonian. The first result improves a previous theorem of Feng and Guo [J.-F. Feng and Y.-B. Guo, Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws, Optimazation57 (2008), no. 1, 135–141]. For the second result, its connectedness condition is sharp since Feng and Guo constructed a 2-connected 1-heavy graph which is almost distance-hereditary but not Hamiltonian.

scholarly journals Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

2014 ◽  
Vol 06 (03) ◽  
pp. 1450043
Author(s):  
Bo Ning ◽  
Shenggui Zhang ◽  
Bing Chen
Keyword(s):  
Hamilton Cycles ◽  
Induced Subgraphs ◽  
Degree Sum ◽  
Degree Condition

Let claw be the graph K1,3. A graph G on n ≥ 3 vertices is called o-heavy if each induced claw of G has a pair of end-vertices with degree sum at least n, and called 1-heavy if at least one end-vertex of each induced claw of G has degree at least n/2. In this note, we show that every 2-connected o-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman and Hedetniemi, Gould and Jacobson, and Shi on the existence of Hamilton cycles in graphs.

scholarly journals A Semiexact Degree Condition for Hamilton Cycles in Digraphs

2010 ◽  
Vol 24 (3) ◽  
pp. 709-756 ◽  
Author(s):  
Demetres Christofides ◽  
Peter Keevash ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Hamilton Cycles ◽  
Degree Condition

scholarly journals A degree condition implying that every matching is contained in a hamiltonian cycle

2009 ◽  
Vol 309 (11) ◽  
pp. 3703-3713 ◽  
Author(s):  
Denise Amar ◽  
Evelyne Flandrin ◽  
Grzegorz Gancarzewicz
Keyword(s):  
Hamiltonian Cycle ◽  
Degree Condition

scholarly journals Long cycles in hypercubes with distant faulty vertices

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Linear Code ◽  
Minimum Distance ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
International Audience ◽  
Long Cycles

Graphs and Algorithms International audience In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.

scholarly journals Multicoloured Hamilton Cycles

10.37236/1204 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Michael Albert ◽  
Alan Frieze ◽  
Bruce Reed
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Proof Technique ◽  
Hamilton Cycles ◽  
Local Lemma

The edges of the complete graph $K_n$ are coloured so that no colour appears more than $\lceil cn\rceil$ times, where $c < 1/32$ is a constant. We show that if $n$ is sufficiently large then there is a Hamiltonian cycle in which each edge is a different colour, thereby proving a 1986 conjecture of Hahn and Thomassen. We prove a similar result for the complete digraph with $c < 1/64$. We also show, by essentially the same technique, that if $t\geq 3$, $c < (2t^2(1+t))^{-1}$, no colour appears more than $\lceil cn\rceil$ times and $t|n$ then the vertices can be partitioned into $n/t$ $t-$sets $K_1,K_2,\ldots,K_{n/t}$ such that the colours of the $n(t-1)/2$ edges contained in the $K_i$'s are distinct. The proof technique follows the lines of Erdős and Spencer's modification of the Local Lemma.

scholarly journals An exact minimum degree condition for Hamilton cycles in oriented graphs

2008 ◽  
Vol 79 (1) ◽  
pp. 144-166 ◽  
Author(s):  
Peter Keevash ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Minimum Degree ◽  
Hamilton Cycles ◽  
Oriented Graphs ◽  
Degree Condition

scholarly journals Uniquely Hamiltonian Characterizations of Distance-Hereditary and Parity Graphs

10.37236/911 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Terry A. McKee
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Odd Order ◽  
Close Relationship

A graph is shown to be distance-hereditary if and only if no induced subgraph of order five or more has a unique hamiltonian cycle; this is also equivalent to every induced subgraph of order five or more having an even number of hamiltonian cycles. Restricting the induced subgraphs to those of odd order five or more gives two similar characterizations of parity graphs. The close relationship between distance-hereditary and parity graphs is unsurprising, but their connection with hamiltonian cycles of induced subgraphs is unexpected.

Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

2021 ◽  
Author(s):  
Dongqin Cheng
Keyword(s):  
Hamiltonian Cycle ◽  
Disjoint Paths ◽  
Hamiltonian Cycles ◽  
Induced Subgraph ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Vertex Disjoint ◽  
Cube Formula ◽  
Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

scholarly journals Degree condition for the existence of a k-factor containing a given Hamiltonian cycle

2009 ◽  
Vol 309 (8) ◽  
pp. 2373-2381 ◽  
Author(s):  
Yunshu Gao ◽  
Guojun Li ◽  
Xuechao Li
Keyword(s):  
Hamiltonian Cycle ◽  
Degree Condition

scholarly journals Ore-degree threshold for the square of a Hamiltonian cycle

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Louis DeBiasio ◽  
Safi Faizullah ◽  
Imdadullah Khan
Keyword(s):  
Graph Theory ◽  
Hamiltonian Cycle ◽  
Blow Up ◽  
Minimum Degree ◽  
Independent Interest ◽  
Regularity Lemma ◽  
Degree Condition ◽  
International Audience ◽  
Classic Theorem

Graph Theory International audience A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.

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