Improved Lower Bounds for the Orders of Even Girth Cages

Jajcayová, Tatiana; Filipovski, Slobodan; Jajcay, Robert

doi:10.37236/6015

Cited by 6 publications

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“…However, the case of even girth remains wide open [4,15]. A well-known conjecture concerning (k, g)cages of even girth asserts that all even girth cages ought to be bipartite (see, for example, [10,13]). In connection to this conjecture, a further specialization of the biregular cage problem has been introduced in [11], in which the authors proposed to study the bipartite biregular (n, m; g)-graphs which are bipartite graphs of even girth g having the degree set {n, m} and satisfying the additional property that the vertices in the same partite set have the same degree.…”

Section: Introductionmentioning

confidence: 99%

Bipartite Biregular Cages and Block Designs

Araujo-Pardo,

Ramos-Rivera,

Jajcay

2019

Preprint

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A bipartite biregular (n, m; g)-graph G is a bipartite graph of even girth g having the degree set {n, m} and satisfying the additional property that the vertices in the same partite set have the same degree. An (n, m; g)bipartite biregular cage is a bipartite biregular (n, m; g)-graph of minimum order. In their 2019 paper, Filipovski, Ramos-Rivera and Jajcay present lower bounds on the orders of bipartite biregular (n, m; g)-graphs, and call the graphs that attain these bounds bipartite biregular Moore cages.In parallel with the well-known classical results relating the existence of k-regular Moore graphs of even girths g = 6, 8 and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of S(2, k, v)-Steiner systems yields the existence of bipartite biregular (k, v−1 k−1 ; 6)-Moore cages. Moreover, in the special case of Steiner triple systems (i.e., in the case k = 3), we completely solve the problem of the existence of (3, m; 6)-bipartite biregular cages for all integers m ≥ 4.Considering girths higher than 6 and prime powers s, we relate the existence of generalized polygons (quadrangles, hexagons and octagons) with the existence of (n + 1, n 2 + 1; 8), (n + 1, n 3 + 1; 12), and (n + 1, n 2 +

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Section: Introductionmentioning

confidence: 99%

Bipartite Biregular Cages and Block Designs

Araujo-Pardo,

Ramos-Rivera,

Jajcay

2019

Preprint

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“…Motivated by the result in Theorem 1.3, which was obtained through counting cycles in a hypothetical graph with given parameters and excess 4, in this paper we address the question of the existence of (k, g)-graphs of excess 4 using spectral properties of their adjacency matrices. The question of the existence of (k, g)-graphs of excess 4 is wide open, and prior to the publication of [11], no such results were known. The results contained in our paper further extend our understanding of the structure of the potential graphs of excess 4.…”

Section: Introductionmentioning

confidence: 99%

“…If J is the all-ones matrix, the sum of the i-distance matrices A i , 0 ≤ i ≤ d, and the matrix E yields d i=0 A i + E = J. To apply the last identity we will use Lemma 4 from [11]. Employing the methodology used by Bannai et al in [1], [2], later by Biggs et al in [3], Delorme et al in [5] and Garbe in [10], we will show that the eigenvalues of G other than ±k are the roots of the polynomials H d−1 (x) + λ.…”

Section: Introductionmentioning

confidence: 99%

“…The following lemma restricts the possible ways in which the four excess vertices are attached to the Moore tree. Lemma 2.1 ( [11]) Let k ≥ 6, g = 2d ≥ 6. Let G be a (k, g)-graph of excess 4, u, v be two adjacent vertices in G, and X f = {w 1 , w 2 , w 3 , w 4 } be the four excess vertices with respect to the edge f = {u, v}.…”

Section: Introductionmentioning

confidence: 99%

“…[11]) Let k ≥ 6, g = 2d ≥ 6. Let G be a (k, g)-graph of excess 4, u, v be two adjacent vertices in G, and X f = {w 1 , w 2 , w 3 , w 4 } be the four excess vertices with respect to the edge f = {u, v}.…”

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On Bipartite Cages of Excess 4

Filipovski

2017

Electron. J. Combin.

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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 via studying spectral properties of their adjacency matrices. We prove that the (k, g)-bipartite graphs of excess 4 satisfy thewhere A denotes the adjacency matrix of the graph in question, J the n × n all-ones matrix, E 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 of degree d − 1. We observe that the eigenvalues other than ±k of these graphs are roots of the polynomials H d−1 (x) + λ, where λ is an eigenvalue of E. Based on the irreducibility of H d−1 (x) ± 2 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 ≥ 6 and k ≡ 1 (mod 3), g = 8, 12, 16 or k ≡ 2 (mod 3), g = 8, and the non-existence of (k, g)-graphs with bicyclic excess 4 if k ≥ 7 is odd number and g = 2d such that d ≥ 4 is even.

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On a relation between bipartite biregular cages, block designs and generalized polygons

Araujo‐Pardo

Jajcay

Ramos-Rivera

et al. 2022

J of Combinatorial Designs

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A bipartite biregular (m,n;g) $(m,n;g)$‐graph normalΓ ${\rm{\Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} $\{m,n\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$‐regular Moore graphs of even girths g=6,8 $g=6,8$, and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $\left(k,\frac{v-1}{k-1};6\right)$‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$‐bipartite biregular cages for all integers n≥4 $n\ge 4$. Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$‐, (n,n+2;8) $(n,n+2;8)$‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.

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Improved Lower Bounds for the Orders of Even Girth Cages

Cited by 6 publications

References 0 publications

Bipartite Biregular Cages and Block Designs

Bipartite Biregular Cages and Block Designs

On Bipartite Cages of Excess 4

On a relation between bipartite biregular cages, block designs and generalized polygons

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