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.

Perfect 1-factorizations

2019 ◽  
Vol 69 (3) ◽  
pp. 479-496 ◽  
Author(s):  
Alexander Rosa
Keyword(s):  
Pairwise Disjoint ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Vertex Set ◽  
Early Survey

AbstractLetGbe a graph with vertex-setV=V(G) and edge-setE=E(G). A 1-factorofG(also calledperfect matching) is a factor ofGof degree 1, that is, a set of pairwise disjoint edges which partitionsV. A 1-factorizationofGis a partition of its edge-setEinto 1-factors. For a graphGto have a 1-factor, |V(G)| must be even, and for a graphGto admit a 1-factorization,Gmust be regular of degreer, 1 ≤r≤ |V| − 1.One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.A 1-factorization ofGis said to beperfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle ofG. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

scholarly journals Toppling numbers of complete and random graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Anthony Bonato ◽  
William B. Kinnersley ◽  
Pawel Pralat
Keyword(s):  
Graph Theory ◽  
Differential Equations ◽  
Random Graphs ◽  
Complete Graph ◽  
Complete Graphs ◽  
Asymptotic Bounds ◽  
Large N ◽  
Chip Firing ◽  
International Audience ◽  
Person Game

Graph Theory International audience We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

scholarly journals Decompositions of Complete Graphs into Bipartite 2-Regular Subgraphs

10.37236/4634 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Darryn Bryant ◽  
Andrea Burgess ◽  
Peter Danziger
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Perfect Matching ◽  
Necessary Conditions ◽  
Complete Graphs ◽  
Edge Disjoint

It is shown that if $G$ is any bipartite 2-regular graph of order at most $n/2$ or at least $n-2$, then the obvious necessary conditions are sufficient for the existence of a decomposition of the complete graph of order $n$ into a perfect matching and edge-disjoint copies of $G$.

scholarly journals On strongly regular graphs with m2 = qm3 and m3 = qm2 where q ∈ Q

2021 ◽  
Vol 109 (123) ◽  
pp. 35-60
Author(s):  
Mirko Lepovic
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Strongly Regular Graph ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular

We say that a regular graph G of order n and degree r ? 1 (which is not the complete graph) is strongly regular if there exist non-negative integers ? and ? such that |Si ? Sj | = ? for any two adjacent vertices i and j, and |Si ? Sj | = ? for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let ?1 = r, ?2 and ?3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, ?2 and ?3, respectively. We here describe the parameters n, r, ? and ? for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 3/2, 4/3, 5/2, 5/3, 5/4, 6/5.

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.

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 Decomposition of (2k+1)-regular graphs containing special spanning 2k-regular Cayley graphs into paths of length 2k+1

2020 ◽  
Author(s):  
Fábio Botler ◽  
Luiz Hoffmann
Keyword(s):  
Regular Graph ◽  
Perfect Matching ◽  
Cayley Graphs ◽  
Regular Graphs

A Pl-decomposition of a graph G is a set of paths with l edges in G that cover the edge set of G. Favaron, Genest, and Kouider (2010) conjectured that every (2k+1)-regular graph that contains a perfect matching admits a P2k+1-decomposition. They also verified this conjecture for 5-regular graphs without cycles of length 4. In 2015, Botler, Mota, and Wakabayashi extended this result to 5-regular graphs without triangles. In this paper, we verify this conjecture for (2k+1)-regular graphs that contain the k-th power of a spanning cycle; and for 5-regular graphs that contain certain spanning 4-regular Cayley graphs.

scholarly journals Gray codes avoiding matchings

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Darko Dimitrov ◽  
Tomáš Dvořák ◽  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Hamiltonian Path ◽  
Gray Code ◽  
Gray Codes ◽  
Similar Characterization ◽  
International Audience ◽  
Forbidden Configurations ◽  
Np Complete ◽  
Necessary And Sufficient

Graphs and Algorithms International audience A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.

scholarly journals Crooked Functions, Bent Functions, and Distance Regular Graphs

10.37236/1372 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
T. D. Bending ◽  
D. Fon-Der-Flaass
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Bent Functions ◽  
Vector Spaces ◽  
Regular Graphs ◽  
Dimensional Vector ◽  
Distance Regular Graph ◽  
Distance Regular Graphs ◽  
Crooked Functions

Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A mapping $Q:V\rightarrow W$ is called crooked if it satisfies the following three properties: $Q(0)=0$; $Q(x)+Q(y)+Q(z)+Q(x+y+z)\neq 0$ for any three distinct $x,y,z$; $Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)\neq 0$ if $a\neq 0$ ($x,y,z$ arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having $\lambda=0$ and $\mu=2$ which is a cover of the complete graph. Our approach is a generalization of a recent construction found by de Caen, Mathon, and Moorhouse. We study graph-theoretical properties of the resulting graphs, including their automorphisms. Also we demonstrate a connection between crooked functions and bent functions.

scholarly journals The Classification of $2$-Extendable Edge-Regular Graphs with Diameter $2$

10.37236/8073 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Klavdija Kutnar ◽  
Dragan Marušič ◽  
Štefko Miklavič ◽  
Primož Šparl
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Perfect Matching ◽  
Negative Integer ◽  
Regular Graphs ◽  
Even Order

Let $\ell$ denote a non-negative integer. A connected graph $\Gamma$ of even order at least $2\ell+2$ is $\ell$-extendable if it contains a matching of size $\ell$ and if every such matching is contained in a perfect matching of $\Gamma$. A connected regular graph $\Gamma$ is edge-regular, if there exists an integer $\lambda$ such that every pair of adjacent vertices of $\Gamma$ have exactly $\lambda$  common neighbours. In this paper we classify $2$-extendable edge-regular graphs of even order and diameter $2$.

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