Regular Graphs Of Girth Research Articles

Small Graphs and Hypergraphs of Given Degree and Girth

The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$. We show that these two problems are in fact very closely linked. By extending the ideas of Cayley graphs to the hypergraph context, we find smallest known hypergraphs for various parameter sets. Because of the close link to the cage problem from graph theory, we are able to use these techniques to find new record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.

Cubic Graphs with Small Independence Ratio

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás proved the upper bound $i(3,\infty) < \frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty) < 0.45537$in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \leqslant 0.454.$

On Bipartite Cages of Excess 4

The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ by studying spectral properties of their adjacency matrices. For a given graph $G$ and for the integers $i$ with $0\leq i\leq diam(G)$, the $i$-distance matrix $A_i$ of $G$ is an $n\times n$ matrix such that the entry in position $(u,v)$ is $1$ if the distance between the vertices $u$ and $v$ is $i$, and zero otherwise. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A=A_{1}$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E=A_{d+1}$ the adjacency matrix of a union of vertex-disjoint cycles, and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter $k-1$ and degree $d-1$. We observe that the eigenvalues other than $\pm k$ of these graphs are roots of the polynomials $H_{d-1}(x)+\lambda$, where $\lambda$ is an eigenvalue of $E$. Based on the irreducibility of $H_{d-1}(x)\pm2$, we give necessary conditions for the existence of these graphs. If $E$ is the adjacency matrix of a cycle of order $n$, we call the corresponding graphs graphs with cyclic excess; if $E$ is the adjacency matrix of a disjoint union of two cycles, we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of $(k,g)$-graphs with cyclic excess $4$ if $k\geq6$ and $k \equiv1 \!\! \pmod {3}$, $g=8, 12, 16$ or $k \equiv2 \!\! \pmod {3}$, $g=8;$ and the non-existence of $(k,g)$-graphs with bicyclic excess $4$ if $k\geq7$ is an odd number and $g=2d$ such that $d\geq4$ is even.

Improved Lower Bounds for the Orders of Even Girth Cages

The well-known Moore bound $M(k,g)$ serves as a universal lower bound for the order of $k$-regular graphs of girth $g$. The excess $e$ of a $k$-regular graph $G$ of girth $g$ and order $n$ is the difference between its order $n$ and the corresponding Moore bound, $e=n - M(k,g) $. We find infinite families of parameters $(k,g)$, $g$ even, for which we show that the excess of any $k$-regular graph of girth $g$ is larger than $4$. This yields new improved lower bounds on the order of $k$-regular graphs of girth $g$ of smallest possible order; the so-called $(k,g)$-cages. We also show that the excess of the smallest $k$-regular graphs of girth $g$ can be arbitrarily large for a restricted family of $(k,g)$-graphs satisfying a very natural additional structural property.

Small Regular Graphs of Girth 7

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.

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

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.

Constructions of biregular cages of girth five

Abstract Let 2 ⩽ 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 extended abstract we construct two infinite families of biregular cages and two semiregular cages, obtained from the incidence graphs of an affine and a biaffine plane by generalizations of 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(18) (2012) 2832–2842].

On regular graphs of girth six arising from projective planes

In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q≥k. Examining the construction method, we prove that starting from PG(2,q), q=ph, p prime, there are no other constructions using this idea resulting in a (q+1−t)-regular graph of girth six than the known ones, if t is not too large (t≤p and roughly t<q1/6/8). Both algebraic and combinatorial tools are used.

Dynamic Cage Survey

A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

Constructions of small regular bipartite graphs of girth 6

In this article, some structures in the projective plane of order $q$ **image** are found which allow us to construct small $k$ **image** -regular balanced bipartite graphs of girth 6 for all $k\le q$ **image** . When $k=q$ **image** , the order of these $q$ **image** -regular graphs is $2(q^2-1)$ **image** ; and when $k\le q-1$ **image** , the order of these $k$ **image** -regular graphs is $2(qk-2)$ **image** . Moreover, the incidence matrix of a $k$ **image** -regular balanced bipartite graph of girth 6 having $2(qk-2)$ **image** vertices, where $k$ **image** is an integer and $q$ **image** is a prime power with $3\le k\le q-1$ **image** , is provided. These graphs improve upon the best known upper bounds for the number of vertices in regular graphs of girth 6. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 121–127 2011 © 2011 Wiley Periodicals, Inc.

On the limitations of the use of solvable groups in Cayley graph cage constructions

A ( k , g ) -cage is a (connected) k -regular graph of girth g having smallest possible order. While many of the best known constructions of small k -regular graphs of girth g are known to be Cayley graphs, there appears to be no general theory of the relationship between the girth of a Cayley graph and the structure of the underlying group. We attempt to fill this gap by focusing on the girth of Cayley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive notion that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girth. Specifically, we establish the existence of upper bounds on the girth of Cayley graphs with respect to the nilpotency class and/or the derived length of the underlying group, when this group is nilpotent or solvable, respectively.

On upper bounds of odd girth cages

We give a construction of k -regular graphs of girth g using only geometrical and combinatorial properties that appear in any ( k ; g + 1 ) -cage, a minimal k -regular graph of girth g + 1 . In this construction, g ≥ 5 and k ≥ 3 are odd integers, in particular when k − 1 is a power of 2 and ( g + 1 ) ∈ { 6 , 8 , 12 } we use the structure of generalized polygons. With this construction we obtain upper bounds for the ( k ; g ) -cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 5 , 7 , 11 .

Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages

Small k -regular graphs of girth g where g = 6 , 8 , 12 are obtained as subgraphs of minimal cages. More precisely, we obtain ( k , 6 ) -graphs on 2 ( k q − 1 ) vertices, ( k , 8 ) -graphs on 2 k ( q 2 − 1 ) vertices and ( k , 12 ) -graphs on 2 k q 2 ( q 2 − 1 ) , where q is a prime power and k is a positive integer such that q ≥ k ≥ 3 . Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 6 , 8 , 12 .

On the number of components of [formula omitted]-cages after vertex deletion

A ( k , g ) -cage is a k -regular graph of girth g and with the least possible number of vertices. In this paper, we investigate the problem of how many connected components there will be after removing a cutset of up to k vertices from a ( k , g ) -cage.

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman–Singleton graph, Ars Combin. 7 (1979) 191–193.] constructed a family of ( p m + 2 ) -regular graphs of girth five and order 2 p 2 m , where p ⩾ 5 is a prime, which includes the Hoffman–Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497–504]. This construction gives an upper bound for the least number f ( k ) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f ( k ) , k ⩾ 16 .

Almost regular cages

Let k be an integer greater than 2. We consider two classes of graphs, which are near to being k-regular. We say that a simple graph is almost regular of Type I if all of its vertices, but one, have degree k, and the exceptional vertex has degree 2. The graph is almost regular of Type II if all of its vertices, but two, have degree k, and the exceptional vertices have degree k − 1. We show that there is a close relationship between the problem of finding small almost regular graphs of girth g and that of finding small (k,g)-graphs (the cage problem).

Some applications of pq-groups in graph theory

We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers. For primes p and q such that p < q and q ≡ 1 (mod p), there exists a nonabelian group of order pq, which is a semidirect product of Zq by Zp [3]. In this note, we use these groups to construct several graphs and graph colorings. Let G be a nonabelian pq-group and let a and b be elements of orders p and q, respectively. The subgroup generated by b is normal in G and a−1ba = b where r ≡ 1 (mod q). So the elements of G can be represented in the form ab , for 0 ≤ i < p and 0 ≤ j < q. Group multiplication can be defined by ab ∗ ab = ab s+t. In what follows, it will be convenient to refer to elements of a pq-group using the notation (i, j) instead of ab . 1 Ramsey Graphs Recall that an (s, t)-coloring of the complete graph Kn is a 2-coloring of its edges such that there is neither a complete subgraph of order s, all of whose edges are color 1, nor a complete subgraph of order t, all of whose edges are color 2. The Ramsey number R(s, t) is the minimum n such that no (s, t) coloring on Kn exists. Our first application of pq-groups gives us a new lower bound for R(4, 8). For this we use a Cayley coloring on the nonabelian group G of order 55, where we take p = 5, q = 11, and r = 3. Note that in a Cayley coloring, we assign

On certain regular graphs of girth 5

Let f(v, 5) be the number of vertices of a (v, 5)‐cage (v ≥ 3). We give an upper bound for f(v, 5) which is considerably better than the previously known upper bounds. In particular, when v = 7, it coincides with the well‐known Hoffman‐ Singleton graph.

The non-existence of certain regular graphs of girth 5

For certain positive integers k it is shown that there is no k-regular graph with girth 5 having k 2 + 3 vertices. This provides a new lower bound for the number of vertices of girth 5 graphs with these valences.

A smallest graph of girth 5 and valency 6

Abstract We give a regular graph of girth 10 and valency 3 which has 70 vertices. We also show that no regular graphs of girth 10 and valency 3 having fewer than 70 vertices exist.