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.

KNOTTED HAMILTONIAN CYCLES IN LINEAR EMBEDDING OF K7 INTO ℝ3

2012 ◽  
Vol 21 (14) ◽  
pp. 1250132 ◽  
Author(s):  
YOUNGSIK HUH
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Intrinsic Properties ◽  
Linear Embedding

In 1983 Conway and Gordon proved that any embedding of the complete graph K7 into ℝ3 contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper, we are interested in knotted Hamiltonian cycles in linear embedding of K7. Concretely it is shown that any linear embedding of K7 contains at most three figure-8 knots.

scholarly journals Fast Strategies in Waiter-Client Games

10.37236/9451 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Dennis Clemens ◽  
Pranshu Gupta ◽  
Fabian Hamann ◽  
Alexander Haupt ◽  
Mirjana Mikalački ◽  
...  
Keyword(s):  
Complete Graph ◽  
Spanning Trees ◽  
Perfect Matchings ◽  
Hamilton Cycles ◽  
The Family ◽  
Winning Set ◽  
Pancyclic Graphs

Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client claims one for himself while the rest go to Waiter. Proceeding like this Waiter wins the game if she forces Client to claim all the elements of any winning set from $\mathcal{F}$. In this paper we study fast strategies for several Waiter-Client games played on the edge set of the complete graph, i.e. $X=E(K_n)$, in which the winning sets are perfect matchings, Hamilton cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.

scholarly journals Alspach's Problem: The Case of Hamilton Cycles and 5-Cycles

10.37236/569 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Heather Jordon
Keyword(s):  
Complete Graph ◽  
Hamilton Cycles

In this paper, we settle Alspach's problem in the case of Hamilton cycles and 5-cycles; that is, we show that for all odd integers $n\ge 5$ and all nonnegative integers $h$ and $t$ with $hn + 5t = n(n-1)/2$, the complete graph $K_n$ decomposes into $h$ Hamilton cycles and $t$ 5-cycles and for all even integers $n \ge 6$ and all nonnegative integers $h$ and $t$ with $hn + 5t = n(n-2)/2$, the complete graph $K_n$ decomposes into $h$ Hamilton cycles, $t$ 5-cycles, and a $1$-factor. We also settle Alspach's problem in the case of Hamilton cycles and 4-cycles.

scholarly journals Sequentially Perfect and Uniform One-Factorizations of the Complete Graph

10.37236/1898 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jeffrey H. Dinitz ◽  
Peter Dukes ◽  
Douglas R. Stinson
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Disjoint Cycles

In this paper, we consider a weakening of the definitions of uniform and perfect one-factorizations of the complete graph. Basically, we want to order the $2n-1$ one-factors of a one-factorization of the complete graph $K_{2n}$ in such a way that the union of any two (cyclically) consecutive one-factors is always isomorphic to the same two-regular graph. This property is termed sequentially uniform; if this two-regular graph is a Hamiltonian cycle, then the property is termed sequentially perfect. We will discuss several methods for constructing sequentially uniform and sequentially perfect one-factorizations. In particular, we prove for any integer $n \geq 1$ that there is a sequentially perfect one-factorization of $K_{2n}$. As well, for any odd integer $m \geq 1$, we prove that there is a sequentially uniform one-factorization of $K_{2^t m}$ of type $(4,4,\dots,4)$ for all integers $t \geq 2 + \lceil \log_2 m \rceil$ (where type $(4,4,\dots,4)$ denotes a two-regular graph consisting of disjoint cycles of length four).

scholarly journals Extending a perfect matching to a Hamiltonian cycle

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Adel Alahmadi ◽  
Robert E. L. Aldred ◽  
Ahmad Alkenani ◽  
Rola Hijazi ◽  
P. Solé ◽  
...  
Keyword(s):  
Graph Theory ◽  
Regular Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Regular Graphs ◽  
Spanning Subgraph ◽  
International Audience ◽  
Matching Extendability

Graph Theory International audience Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

Cyclic near-Hamiltonian cycle decomposition of 2-fold complete graph

2019 ◽  
Author(s):  
Raja’i Aldiabat ◽  
Haslinda Ibrahim ◽  
Sharmila Karim
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Cycle Decomposition

scholarly journals Cyclic Hamiltonian cycle systems of the complete graph

2004 ◽  
Vol 279 (1-3) ◽  
pp. 107-119 ◽  
Author(s):  
Marco Buratti ◽  
Alberto Del Fra
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Cycle Systems

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 On Two Problems Regarding the Hamiltonian Cycle Game

10.37236/117 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Dan Hefetz ◽  
Sebastian Stich
Keyword(s):  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Winning Strategy ◽  
Minimum Number

We consider the fair Hamiltonian cycle Maker-Breaker game, played on the edge set of the complete graph $K_n$ on $n$ vertices. It is known that Maker wins this game if $n$ is sufficiently large. We are interested in the minimum number of moves needed for Maker in order to win the Hamiltonian cycle game, and in the smallest $n$ for which Maker has a winning strategy for this game. We prove the following results: (1) If $n$ is sufficiently large, then Maker can win the Hamiltonian cycle game within $n+1$ moves. This bound is best possible and it settles a question of Hefetz, Krivelevich, Stojaković and Szabó; (2) If $n \geq 29$, then Maker can win the Hamiltonian cycle game. This improves the previously best bound of $600$ due to Papaioannou.

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