scholarly journals Small Regular Graphs of Girth 7

10.37236/4205 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
M. Abreu ◽  
G. Araujo-Pardo ◽  
C. Balbuena ◽  
D. Labbate ◽  
J. Salas
Keyword(s):  
Prime Power ◽  
Minimum Degree ◽  
Regular Graphs ◽  
Geometric Properties

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$. A corrigendum was added to this paper on 21 June 2016.

scholarly journals A Construction of Small $(q-1)$-Regular Graphs of Girth 8

10.37236/4397 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
M. Abreu ◽  
G. Araujo-Pardo ◽  
C. Balbuena ◽  
D. Labbate
Keyword(s):  
Prime Power ◽  
Infinite Family ◽  
Regular Graphs

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth 8 and order $2q(q-1)^2$ for all prime powers $q\geq 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.

scholarly journals Distant Set Distinguishing Total Colourings of Graphs

10.37236/5481 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Method ◽  
Constant Factor ◽  
Additive Constant ◽  
Regular Graphs

The Total Colouring Conjecture suggests that $\Delta+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $\Delta$. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in $G$ to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form $\Delta+C$, for a constant $C>0$ still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most $r$, if only $G$ has minimum degree $\delta$ larger than a constant dependent on $r$. We prove that such assumption on $\delta$ is then unavoidable and exploit the probabilistic method in order to provide two supporting results for the conjecture. Namely, we prove the upper bound $(1+o(1))\Delta$ for every $r$, and show that for any fixed $\epsilon\in(0,1]$ and $r$, the conjecture holds if $\delta\geq \varepsilon\Delta$, i.e., in particular for regular graphs.

scholarly journals Phase transition in the bipartite z-matching

2021 ◽  
Vol 94 (12) ◽  
Author(s):  
Till Kahlke ◽  
Martin Fränzle ◽  
Alexander K. Hartmann
Keyword(s):  
Random Graphs ◽  
Minimum Degree ◽  
Finite Size ◽  
Exact Algorithm ◽  
Regular Graphs ◽  
Maximum Cardinality ◽  
Exact Matching ◽  
Matching Algorithm ◽  
Matching Problems ◽  
Strong Change

Abstract We study numerically the maximum z-matching problems on ensembles of bipartite random graphs. The z-matching problems describes the matching between two types of nodes, users and servers, where each server may serve up to z users at the same time. Using a mapping to standard maximum-cardinality matching, and because for the latter there exists a polynomial-time exact algorithm, we can study large system sizes of up to $$10^6$$ 10 6 nodes. We measure the capacity and the energy of the resulting optimum matchings. First, we confirm previous analytical results for bipartite regular graphs. Next, we study the finite-size behaviour of the matching capacity and find the same scaling behaviour as before for standard matching, which indicates the universality of the problem. Finally, we investigate for bipartite Erdős–Rényi random graphs the saturability as a function of the average degree, i.e. whether the network allows as many customers as possible to be served, i.e. exploiting the servers in an optimal way. We find phase transitions between unsaturable and saturable phases. These coincide with a strong change of the running time of the exact matching algorithm, as well with the point where a minimum-degree heuristic algorithm starts to fail. Graphical Abstract

scholarly journals Optimal Packings of Hamilton Cycles in Graphs of High Minimum Degree

2012 ◽  
Vol 22 (3) ◽  
pp. 394-416 ◽  
Author(s):  
DANIELA KÜHN ◽  
JOHN LAPINSKAS ◽  
DERYK OSTHUS
Keyword(s):  
General Result ◽  
Minimum Degree ◽  
Regular Graphs ◽  
Hamilton Cycles ◽  
Edge Disjoint ◽  
Large Graph ◽  
Spanning Subgraph ◽  
Hamilton Decomposition

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ=(1/2+α)n. For any constant α>0, we give an optimal answer in the following sense: let regeven(n,δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals regeven(n,δ)/2. The value of regeven(n,δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

scholarly journals Biregular Cages of Girth Five

10.37236/2594 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Marién Abreu ◽  
Gabriela Araujo-Pardo ◽  
Camino Balbuena ◽  
Domenico Labbate ◽  
Gloria López-Chávez
Keyword(s):  
Prime Power ◽  
Discrete Math ◽  
Regular Graphs ◽  
Minimum Order ◽  
Degree Set ◽  
Positive Integers

Let $2 \le r < m$ and $g$ be positive integers. An $(\{r,m\};g)$-graph (or biregular graph) is a graph with degree set $\{r,m\}$ and girth $g$, and an $(\{r,m\};g)$-cage (or biregular cage) is an $(\{r,m\};g)$-graph of minimum order $n(\{r,m\};g)$. If $m=r+1$, an $(\{r,m\};g)$-cage is said to be a semiregular cage.In this paper we generalize the reduction and graph amalgam operations from [M. Abreu,  G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth $5$. Discrete Math. 312 (2012) 2832--2842] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are $(\{r,2r-3\};5)$-cages for all $r=q+1$ with $q$ a prime power, and $(\{r,2r-5\};5)$-cages for all $r=q+1$ with $q$ a prime. The new semiregular cages are constructed for $r=5$ and $6$ with $31$ and $43$ vertices respectively.

scholarly journals Independent Sets in Graphs with Given Minimum Degree

10.37236/2722 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
James Alexander ◽  
Jonathan Cutler ◽  
Tim Mink
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Independent Sets ◽  
Regular Graphs ◽  
Double Covers ◽  
Delta G ◽  
Regular Bipartite Graph

The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late.  Let $i(G)$ be the number of independent sets in a graph $G$ and let $i_t(G)$ be the number of independent sets in $G$ of size $t$.  Kahn used entropy to show that if $G$ is an $r$-regular bipartite graph with $n$ vertices, then $i(G)\leq i(K_{r,r})^{n/2r}$.  Zhao used bipartite double covers to extend this bound to general $r$-regular graphs.  Galvin proved that if $G$ is a graph with $\delta(G)\geq \delta$ and $n$ large enough, then $i(G)\leq i(K_{\delta,n-\delta})$.  In this paper, we prove that if $G$ is a bipartite graph on $n$ vertices with $\delta(G)\geq\delta$ where $n\geq 2\delta$, then $i_t(G)\leq i_t(K_{\delta,n-\delta})$ when $t\geq 3$.  We note that this result cannot be extended to $t=2$ (and is trivial for $t=0,1$).  Also, we use Kahn's entropy argument and Zhao's extension to prove that if $G$ is a graph with $n$ vertices, $\delta(G)\geq\delta$, and $\Delta(G)\leq \Delta$, then $i(G)\leq i(K_{\delta,\Delta})^{n/2\delta}$.

Excluding Minors in Nonplanar Graphs of Girth at Least Five

2000 ◽  
Vol 9 (6) ◽  
pp. 573-585 ◽  
Author(s):  
ROBIN THOMAS ◽  
JAN McDONALD THOMSON
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
The Other ◽  
Petersen Graph ◽  
Regular Graphs ◽  
Nonplanar Graph ◽  
A Minor

A graph G is quasi 4-connected if it is simple, 3-connected, has at least five vertices, and for every partition (A, B, C) of V(G) either [mid ]C[mid ] [ges ] 4, or G has an edge with one end in A and the other end in B, or one of A,B has at most one vertex. We show that any quasi 4-connected nonplanar graph with minimum degree at least three and no cycle of length less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result of Kilakos and Shepherd who proved the same for 3-regular graphs.

scholarly journals Dirac’s theorem for random regular graphs

2020 ◽  
pp. 1-20
Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
António Girão ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Regular Graph ◽  
Minimum Degree ◽  
Regular Graphs ◽  
Degree Bound

Abstract We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to $\epsilon > 0$ , a.a.s. the following holds. Let $G'$ be any subgraph of the random n-vertex d-regular graph $G_{n,d}$ with minimum degree at least $$(1/2 + \epsilon )d$$ . Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.

scholarly journals A Prolific Construction of Strongly Regular Graphs with the $n$-e.c. Property

10.37236/1647 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Dudley Stark
Keyword(s):  
Regular Graph ◽  
Prime Power ◽  
Probabilistic Methods ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Existentially Closed ◽  
Large N ◽  
Vertex Set ◽  
Strongly Regular ◽  
Hadamard Designs

A graph is $n$-e.c.$\,$ ($n$-existentially closed) if for every pair of subsets $U$, $W$ of the vertex set $V$ of the graph such that $U\cap W=\emptyset$ and $|U|+|W|=n$, there is a vertex $v\in V-(U\cup W)$ such that all edges between $v$ and $U$ are present and no edges between $v$ and $W$ are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices $v_1,v_2\in V$ depends only on whether or not $\{v_1,v_2\}$ is an edge in the graph. The only strongly regular graphs that are known to be $n$-e.c. for large $n$ are the Paley graphs. Recently D. G. Fon-Der-Flaass has found prolific constructions of strongly regular graphs using affine designs. He notes that some of these constructions were also studied by Wallis. By taking the affine designs to be Hadamard designs obtained from Paley tournaments, we use probabilistic methods to show that many non-isomorphic strongly regular $n$-e.c. graphs of order $(q+1)^2$ exist whenever $q\geq 16 n^2 2^{2n}$ is a prime power such that $q\equiv 3\!\!\!\pmod{4}$.

scholarly journals Perfect Matchings in $\epsilon$-regular Graphs

10.37236/1351 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Noga Alon ◽  
Vojtech Rödl ◽  
Andrzej Ruciński
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev.

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