scholarly journals Rainbow Pancyclicity in Graph Systems

10.37236/9033 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Yangyang Cheng ◽  
Guanghui Wang ◽  
Yi Zhao
Keyword(s):  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Main Tool ◽  
Vertex Set ◽  
Absorption Technique

 Let $G_1,\ldots,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set $\{e_1,\ldots,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can be viewed as a rainbow version of the well-known Dirac theorem. In this paper, we prove this conjecture asymptotically by showing that for every $\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any graphs $G_1,\ldots,G_n$ on the same vertex set of size $n$ with $\delta(G_i)\ge (\frac{1}{2}+\varepsilon)n$, there exists a rainbow Hamiltonian cycle. Our main tool is the absorption technique. Additionally, we prove that with $\delta(G_i)\geq \frac{n+1}{2}$ for each $i$, one can find rainbow cycles of length $3,\ldots,n-1$.

scholarly journals The Bandwidth Theorem in sparse graphs

2020 ◽  
Author(s):  
Peter Allen ◽  
Julia Böttcher ◽  
Julia Ehrenmüller ◽  
Anusch Taraz
Keyword(s):  
Hamiltonian Cycle ◽  
Blow Up ◽  
Minimum Degree ◽  
Main Tool ◽  
Sparse Graphs ◽  
Spanning Subgraphs ◽  
Dense Graphs ◽  
Vertex Graph ◽  
First Results ◽  
Delta G

One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $\left( \frac{r-1}{r} + o(1)\right) n$ contains a copy of all $n$-vertex graphs $H$ such that $\chi(H) \leq r$, $\Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $\Delta$, if $p \gg \left(\frac{\log{n}}{n}\right)^{1/\Delta}$, then asymptotically almost surely, every subgraph $G\subseteq G(n, p)$ with $\delta(G) \geq \left( \frac{r-1}{r} + o(1)\right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $\Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.

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.

Fault-free Hamiltonian cycle including given edges in folded hypercubes with faulty edges

2020 ◽  
Vol 12 (03) ◽  
pp. 2050036
Author(s):  
Dongqin Cheng
Keyword(s):  
Hamiltonian Cycle ◽  
Interconnection Network ◽  
Free Edge ◽  
Multiprocessor Systems ◽  
Linear Forest ◽  
Vertex Set ◽  
Vertex Disjoint ◽  
Folded Hypercube

The folded hypercube is an important interconnection network for multiprocessor systems. Let [Formula: see text] with [Formula: see text] denote an [Formula: see text]-dimensional folded hypercube. For a given fault-free edge set [Formula: see text] with [Formula: see text] and a faulty edge set [Formula: see text] with [Formula: see text], in this paper we prove that [Formula: see text] contains a fault-free Hamiltonian cycle including each edge of [Formula: see text] if and only if the subgraph induced by [Formula: see text] is linear forest. Furthermore, we give the definitions of the distance among three vertex-disjoint edges and the distance between a vertex and a vertex set. For three vertex-disjoint edges [Formula: see text], the distance among them is denoted by [Formula: see text]. For a vertex [Formula: see text] and a vertex set [Formula: see text], the distance between [Formula: see text] and [Formula: see text] is denoted by [Formula: see text].

scholarly journals On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

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 Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

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.

Degree Kirchhoff Index of Bicyclic Graphs

2017 ◽  
Vol 60 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Zikai Tang ◽  
Hanyuan Deng
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Maximum And Minimum Degree ◽  
Bicyclic Graphs

AbstractLet G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

Dominating a Family of Graphs with Small Connected Subgraphs

2000 ◽  
Vol 9 (4) ◽  
pp. 309-313 ◽  
Author(s):  
YAIR CARO ◽  
RAPHAEL YUSTER
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Minimum Size ◽  
Dominating Sets ◽  
Asymptotic Results ◽  
Connected Dominating Sets ◽  
Vertex Set ◽  
Connected Subgraphs

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

scholarly journals The H-force sets of the graphs satisfying the condition of Ore’s theorem

2020 ◽  
Vol 18 (1) ◽  
pp. 771-780
Author(s):  
Xinhong Zhang ◽  
Ruijuan Li
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Graph ◽  
Vertex Graph ◽  
Vertex Set

Abstract Let G be a Hamiltonian graph. A nonempty vertex set X\subseteq V(G) is called a Hamiltonian cycle enforcing set (in short, an H-force set) of G if every X-cycle of G (i.e., a cycle of G containing all vertices of X) is a Hamiltonian cycle. For the graph G, h(G) (called the H-force number of G) is the smallest cardinality of an H-force set of G. Ore’s theorem states that an n-vertex graph G is Hamiltonian if d(u)+d(v)\ge n for every pair of nonadjacent vertices u,v of G. In this article, we study the H-force sets of the graphs satisfying the condition of Ore’s theorem, show that the H-force number of these graphs is possibly n, or n-2 , or \frac{n}{2} and give a classification of these graphs due to the H-force number.

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