scholarly journals Perfect 1-Factorisations of Circulants with Small Degree

10.37236/2264 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Sarada Herke ◽  
Barbara Maenhaut
Keyword(s):  
Large Family ◽  
Small Degree ◽  
Hamilton Cycle ◽  
Perfect Matchings ◽  
Existence Problem ◽  
Circulant Graphs ◽  
Edge Disjoint ◽  
Even Order

A $1$-factorisation of a graph $G$ is a decomposition of $G$ into edge-disjoint $1$-factors (perfect matchings), and a perfect $1$-factorisation is a $1$-factorisation in which the union of any two of the $1$-factors is a Hamilton cycle.  We consider the problem of the existence of perfect $1$-factorisations of even order circulant graphs with small degree.  In particular, we characterise the $3$-regular circulant graphs that admit a perfect $1$-factorisation and we solve the existence problem for a large family of $4$-regular circulants.  Results of computer searches for perfect  $1$-factorisations of $4$-regular circulant graphs of orders up to $30$ are provided and some problems are posed.

scholarly journals Pretty Good State Transfer on Circulant Graphs

10.37236/6388 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Hiranmoy Pal ◽  
Bikash Bhattacharjya
Keyword(s):  
Complex Number ◽  
Adjacency Matrix ◽  
Transition Matrix ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Edge Disjoint ◽  
State Transfer ◽  
Good State

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

scholarly journals An approximate version of Jackson’s conjecture

2020 ◽  
Vol 29 (6) ◽  
pp. 886-899
Author(s):  
Anita Liebenau ◽  
Yanitsa Pehova
Keyword(s):  
Bipartite Graph ◽  
Small Proportion ◽  
Complete Bipartite Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Bipartite Digraph ◽  
Bipartite Tournament ◽  
Approximate Version

AbstractA diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and out-degree. In 1981 Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact its edge set can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every ε > 0 there exists n0 such that every diregular bipartite tournament on 2n ≥ n0 vertices contains a collection of (1/2–ε)n cycles of length at least (2–ε)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c > 1/2 and ε > 0 there exists n0 such that every cn-regular bipartite digraph on 2n ≥ n0 vertices contains (1−ε)cn edge-disjoint Hamilton cycles.

scholarly journals Edge Disjoint Hamilton Cycles in Knödel Graphs

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Palanivel Subramania Nadar Paulraja ◽  
S Sampath Kumar
Keyword(s):  
Maximum Degree ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
Cycle Decomposition ◽  
Edge Disjoint ◽  
International Audience ◽  
Appl Math

International audience The vertices of the Knödel graph $W_{\Delta, n}$ on $n \geq 2$ vertices, $n$ even, and of maximum degree $\Delta, 1 \leq \Delta \leq \lfloor log_2(n) \rfloor$, are the pairs $(i,j)$ with $i=1,2$ and $0 \leq j \leq \frac{n}{2} -1$. For $0 \leq j \leq \frac{n}{2} -1$, there is an edge between vertex $(1,j)$ and every vertex $(2,j + 2^k - 1 (mod \frac{n}{2}))$, for $k=0,1,2, \ldots , \Delta -1$. Existence of a Hamilton cycle decomposition of $W_{k, 2k}, k \geq 6$ is not yet known, see Discrete Appl. Math. 137 (2004) 173-195. In this paper, it is shown that the $k$-regular Knödel graph $W_{k,2k}, k \geq 6$ has $ \lfloor \frac{k}{2} \rfloor - 1$ edge disjoint Hamilton cycles.

scholarly journals Random Graph's Hamiltonicity is Strongly Tied to its Minimum Degree

10.37236/8339 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Yahav Alon ◽  
Michael Krivelevich
Keyword(s):  
Random Graph ◽  
Analogous Result ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Perfect Matchings ◽  
Delta G

We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.

scholarly journals A Formula for the Energy of Circulant Graphs with Two Generators

2016 ◽  
Vol 2016 ◽  
pp. 1-4
Author(s):  
Justine Louis
Keyword(s):  
Complete Graph ◽  
Hamilton Cycle ◽  
Circulant Graphs

We derive closed formulas for the energy of circulant graphs generated by1andγ, whereγ⩾2is an integer. We also find a formula for the energy of the complete graph without a Hamilton cycle.

scholarly journals FROBENIUS CIRCULANT GRAPHS OF VALENCY FOUR

2008 ◽  
Vol 85 (2) ◽  
pp. 269-282 ◽  
Author(s):  
ALISON THOMSON ◽  
SANMING ZHOU
Keyword(s):  
Cayley Graph ◽  
Cyclic Group ◽  
Interconnection Networks ◽  
Frobenius Group ◽  
Optimal Routing ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Frobenius Kernel ◽  
Even Order

AbstractA first kind Frobenius graph is a Cayley graph Cay(K,S) on the Frobenius kernel of a Frobenius group $K \rtimes H$ such that S=aH for some a∈K with 〈aH〉=K, where H is of even order or a is an involution. It is known that such graphs admit ‘perfect’ routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.

scholarly journals Restricted triangulation on circulant graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 358-369 ◽  
Author(s):  
Niran Abbas Ali ◽  
Adem Kilicman ◽  
Hazim Michman Trao
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Problem ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Value Set ◽  
Vertex Set ◽  
Necessary And Sufficient

AbstractThe restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C(n, S) be a circulant graph on n vertices with jump value set S. We consider the restricted triangulation existence problem for G. We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S(n) that has the smallest cardinality with C(n, S(n)) admits a restricted triangulation. We present the measure of non-triangulability of Kn − G for a given G.

scholarly journals Subgraphs of Randomk-Edge-Colouredk-Regular Graphs

2009 ◽  
Vol 18 (4) ◽  
pp. 533-549 ◽  
Author(s):  
PAULETTE LIEBY ◽  
BRENDAN D. McKAY ◽  
JEANETTE C. McLEOD ◽  
IAN M. WANLESS
Keyword(s):  
Regular Graph ◽  
Asymptotic Estimate ◽  
Forthcoming Paper ◽  
Perfect Matchings ◽  
Regular Graphs ◽  
Random Set ◽  
Expected Number ◽  
Edge Disjoint ◽  
Edge Colouring

LetG=G(n) be a randomly chosenk-edge-colouredk-regular graph with 2nvertices, wherek=k(n). Such a graph can be obtained from a random set ofkedge-disjoint perfect matchings ofK2n. Leth=h(n) be a graph withm=m(n) edges such thatm2+mk=o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs ofGisomorphic toh. Isomorphisms may or may not respect the edge colouring, and other generalizations are also presented. Special attention is paid to matchings and cycles.The results in this paper are essential to a forthcoming paper of McLeod in which an asymptotic estimate for the number ofk-edge-colouredk-regular graphs fork=o(n5/6) is found.

scholarly journals Vertex-Oriented Hamilton Cycles in Directed Graphs

10.37236/204 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael J. Plantholt ◽  
Shailesh K. Tipnis
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Sufficient Conditions ◽  
Hamilton Cycle ◽  
Directed Path ◽  
Hamilton Cycles ◽  
Even Order

Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

scholarly journals The Degree-Diameter Problem for Circulant Graphs of Degree 8 and 9

10.37236/4279 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Robert R. Lewis
Keyword(s):  
Lower Bounds ◽  
Small Degree ◽  
Extremal Graphs ◽  
Circulant Graphs

This paper considers the degree-diameter problem for undirected circulant graphs. The focus is on extremal graphs of given (small) degree and arbitrary diameter. The published literature only covers graphs of up to degree 7. The approach used to establish the results for degree 6 and 7 has been extended successfully to degree 8 and 9. Candidate graphs are defined as functions of the diameter for both degree 8 and degree 9. They are proven to be extremal for small diameters. They establish new lower bounds for all greater diameters, and are conjectured to be extremal. The existence of the degree 8 solution is proved for all diameters. Finally some conjectures are made about solutions for circulant graphs of higher degree.

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