M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate
In this paper we obtain $(q+3)$--regular graphs of girth 5 with fewer vertices than previously known ones for $q=13,17,19$ and for any prime $q \ge 23$ performing operations of reductions and amalgams on the Levi graph $B_q$ of an elliptic semiplane of type ${\cal C}$. We also obtain a 13-regular graph of girth 5 on 236 vertices from $B_{11}$ using the same technique.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The first known families of cages arised from the incidence graphs of generalized polygons of order $q$, $q$ a prime power. In particular, $(q+1,6)$--cages have been obtained from the projective planes of order $q$. Morever, infinite families of small regular graphs of girth 5 have been constructed performing algebraic operations on $\mathbb{F}_q$. In this paper, we introduce some combinatorial operations to construct new infinite families of small regular graphs of girth 7 from the $(q+1,8)$--cages arising from the generalized quadrangles of order $q$, $q$ a prime power.
In this paper we are interested in the {\it{Cage Problem}} that consists in constructing regular graphs of given girth $g$ and minimum order. We focus on girth $g=5$, where cages are known only for degrees $k \le 7$. We construct regular graphs of girth $5$ using techniques exposed by Funk [Note di Matematica. 29 suppl.1, (2009) 91 - 114] and Abreu et al. [Discrete Math. 312 (2012), 2832 - 2842] to obtain the best upper bounds known hitherto. The tables given in the introduction show the improvements obtained with our results.
On edge-girth-regular graphs: lower bounds and new families
本文利用广义四边形和椭圆二次曲面铅笔构造新的边围长正则图族，并借助邻接矩阵特征值性质改进极值边围长正则图的阶下界。
We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the edges with end-vertex $u$ lie on. Girth-regularity generalises two very different aspects of symmetry in graph theory: that of vertex transitivity and that of distance-regularity. For general girth-regular graphs, we give some results on the extremal cases of signatures. We then focus on the cubic case and provide a characterisation of cubic girth-regular g…
This paper presents a characterization of edge-transitive graphs which are four regular and have girth four. This class consists of four infinite families plus four exceptional graphs.
It is well known that 3--regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3--regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1--, 2-- or 3--edge-connected depending on the construction chosen. From the constructions arise (naive) upper bounds on the size of the smallest non-Hamiltonian 3--regular graphs with particular girth. Several examples are given of the smallest such graphs for various choices of girth and connecte…
Tight factorizations of girth-$g$-regular graphs
研究围长、正则度和色指数相等的围长正则图，确定每个1-因子均与所有围长圈相交的1-因子分解，并探讨其在哈密顿可分解性、三维几何及优化问题中的应用。
Geometric constructions of small regular graphs with girth 7
该研究利用广义四边形的关联图进行几何构造，得到围长为7的三类正则图无限族，部分图达到目前已知最小阶数。
On strongly regular signed graphs of higher girth
研究高围长的强正则带号图，证明围长为4的二分图由加权矩阵设计和对称块设计分类，围长为5的易于描述，围长更高的不存在，围长为4的非二分图未解决。
On vertex-girth-regular graphs: (Non-)existence, bounds and enumeration
本文研究了顶点围长正则图的基本性质，给出了参数需满足的存在关系、无限族构造及顶点数下界，并计算了小围长最小三次和四次图的阶数。
We consider 3-regular planar matchstick graphs, i.e. those which have a planar embedding such that all edge lengths are equal, with given girth g. For girth 3 it is known that such graphs exist if and only if the number of vertices n is an even integer larger or equal to 8. Here we prove that such graphs exist for girth g=4 if and only if n is even and at least 20. We provide an example for girth g=5 consisting of 180 vertices.
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 $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.
In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its geometrical correctness.
Nonexistence of certain edge-girth-regular graphs
本文证明了围长为 7 或 8 的两类 3-正则边围长正则图不存在，证实了 Goedgebeur 和 Jooken 的猜想，并解决了若干其他未决情形。
Larger matchings and independent sets in regular uniform hypergraphs of high girth
本文分析了两种算法，分别用于在大围长的 $k$-一致 $d$-正则超图上生成匹配和独立集，从而获得了此类超图中最大匹配或独立集大小的新下界。
For every integer d > 9, we construct infinite families {G_n}_n of d+1-regular graphs which have a large girth > log_d |G_n|, and for d large enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I_n}_n of d+1-…
A non-existence result for vertex-girth-regular graphs
本文证明了对于k≥3和ε≤(k-1)/2，不存在vgr(n,k,5, [k(k-1)^2/2]-ε)图，填补了圈长为5时的非存在性空白。
We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices. As long as the smallest degree $δ(G)<k$, choose, uniformly at random, two vertices $u,v \in V(G)$ of degree $δ(G)$ whose distance is at least $g-1$. If there are no such vertex pairs, abort. Otherwise, add the edge $uv$ to $E(G)$. We show that with high probability this algorithm yields a $k$-regular graph with girth at least $g$. Our analysis also implies …
An edge-girth-regular graph $egr(v,k,g,λ)$, is a $k$-regular graph of order $v$, girth $g$ and with the property that each of its edges is contained in exactly $λ$ distinct $g$-cycles. An $egr(v,k,g,λ)$ is called extremal for the triple $(k,g,λ)$ if $v$ is the smallest order of any $egr(v,k,g,λ)$. In this paper, we introduce two families of edge-girth-regular graphs. The first one is a family of extremal $egr(2q^2,q,6,(q-1)^2(q-2))$ for any prime power $q\geq 3$ and, the second one is a family of $egr(q(q^2+1),q,5,λ)$ for $λ\geq q-1$ and $q\geq 8$ an odd power of $2$. In particular, if $q=8$ w…
On extremal (almost) edge-girth-regular graphs
该研究聚焦边围长正则图（egr-图）的极值问题，证明有限平面关联图衍生的部分图为三元组$(k,g,λ)$的极值egr-图，并给出egr-图阶数的新下界。
An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.
We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform `Cayley' hypergraph on $S_n$ which has girth $Ω(n^{1/3})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.
We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This is an uncommon direction of transfer of results, which is usually from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a p…
Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The first contribution of this paper is a construction of these graphs in an alternative way by means of an explicit formula using graphical terminology. Furthermore by removing some specific perfect dominating sets from a $(q+1,8)$-cage we derive $k$-regular graphs of girth 8 for $k= q-1$ and $k=q$, having the smallest number of vertices known so far.
A classification of $Q$-polynomial distance-regular graphs with girth $6$
本文分类了围长达到上界 $6$ 的 $Q$-多项式距离正则图，证明其同构于奇图或广义六边形。
The strong spectral property for some families of unicyclic graphs
本文研究具有强谱性质的图类，完全刻画了围长为三的单圈图，并证明围长至多五的蝌蚪图具有该性质而围长六的蝌蚪图不具有。
Semicubic cages and small graphs of even girth from voltage graphs
本文针对偶围长小阶半立方图的构造问题，采用粘合立方笼的推广方法与基于$\boldsymbol{\rm Z}_{m}$电压图的核心技术，构造了多族不同偶围长的半立方图，部分实例达到已知最优界。
In this paper, we consider a randomized greedy algorithm for independent sets in $r$-uniform $d$-regular hypergraphs $G$ on $n$ vertices with girth $g$. By analyzing the expected size of the independent sets generated by this algorithm, we show that $α(G)\geq (f(d,r)-ε(g,d,r))n$, where $ε(g,d,r)$ converges to $0$ as $g\rightarrow\infty$ for fixed $d$ and $r$, and $f(d,r)$ is determined by a differential equation. This extends earlier results of Gamarnik and Goldberg for graphs. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the in…
We show that the abelian girth of a graph is at least three times its girth. We prove an analogue of the Moore bound for the abelian girth of regular graphs, where the degree of the graph is fixed and the number of vertices is large. We conclude that one could try to improve the Moore bound for graphs of fixed degree and many vertices by trying to improve its analogue concerning the abelian girth.
Recently, it has been shown that a connected graph $Γ$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance-regular graphs in terms of the predistance polynomial of degree $d$.
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 (2011) 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…
The main purpose of this paper is to construct high-girth regular expander graphs with localized eigenvectors for general degrees, which is inspired by a recent work due to Alon, Ganguly and Srivastava (to appear in Israel J. Math.).
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…
We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather than quaternions. A key tool is the existing result of the unique factorization of integral octonions. The families obtained by this mean present not only the same spectral property that make them good expanders, but also show a larger girth, yielding a new record for regular graphs.
We show that if $n=7k/i$ with $i \in \{1,2,3\}$ then the cop number of the generalised Petersen graph $GP(n,k)$ is $4$, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most $4$. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number $4$ but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised P…
We find new upper bounds on the size of a minimum totally dominating set for random regular graphs and for regular graphs with large girth. These bounds are obtained through the analysis of a local algorithm using a method due to Hoppen and Wormald [Local algorithms, regular graphs of large girth, and random regular graphs. Combinatorica 38(3) (2018), 619-664].
Total coloring of regular graphs of girth = degree + 1
本文研究围长等于度数加1的正则图的高效全染色，构造了围长为4的三次图的高效全染色并提出四种基本运算猜想，同时给出若干非高效全染色实例及对3-路和3-星划分的应用。
Suzuki (2004) [7] classified thin weakly distance-regular digraphs and pro- posed the project to classify weakly distance-regular digraphs of valency 3. The case of girth 2 was classified by the third author (2004) [9] under the assumption of the commutativity. In this paper, we continue this project and classify these digraphs with girth more than 2 and two types of arcs.
The edge-girth-regularity of Wenger graphs
本文证明 Wenger 图与线性化 Wenger 图为边围长正则图并确定其参数，据此推出围长环数、广义 Turán 数下界及奇数 q 下的极值渐近结论。
In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth.
On the Girth of Graph Lifts
本文研究了图提升的圈长问题，提出了新的函数n(H,g)并证明了其与经典界的关系，同时探讨了度分布相关的推广问题。
We present a local algorithm producing an independent set of expected size $0.44533n$ on large-girth 3-regular graphs and $0.40407n$ on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with $1.34105n$ edges on large-girth 3-regular graphs. These decrease the gaps between the best known upper and lower bounds from $0.0178$ to $0.01$, from $0.0242$ to $0.0123$ and from $0.0724$ to $0.0616$, respectively. We are using local algorithms, therefore, the method also provides upper bounds for the fractional coloring numbers of $1 / 0.44533 \approx 2.24554$ and…
Exhaustive generation of edge-girth-regular graphs
本文研究边-围长正则图的最小阶数$n(k,g,λ)$，提出线性时间计数算法和穷举生成算法，通过大规模计算得到多个新的极值图并改进上下界，同时否证了相关猜想。
We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that $|λ| \leq 2\sqrt{d-1}$. We then use these approximations for $λ= -2\sqrt{d-1}$ to produce factor of i.i.d. ind…
On tetravalent half-arc-transitive graphs of girth 5
本文研究围长为5的四度半弧传递图，证明除两个例外外其5-圈均为有向且一致，并给出无限族及紧附着图与弱亚循环图的分类。
For integer $k\geq2$ and prime power $q$, the algebraic bipartite graph $D(k,q)$ proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and cryptography. This graph is $q$-regular, edge-transitive and of girth at least $k+4$. For its exact girth $g=g(D(k,q))$, Füredi et al. (1995) conjectured $g=k+5$ for odd $k$ and $q\geq4$. This conjecture was shown to be valid in 2016 when $(k+5)/2$ is the product of an arbitrary factor of $q-1$ and an arbitrary power of the characteristic of $\mathbb{F}_q$. In this paper, we determine all the gir…
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minim…
We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. Typical processes are defined in a way that they create a correspondence principle between random $d$-reguar graphs and ergodic theory on $T_d$. Using this correspondence principle together …
Cubic vertex-transitive graphs of girth six
本文完整分类了围长为6的有限简单三次点传递图，证明除Desargues图外，它们均可由环面六边形平铺、闭双曲曲面弧传递三角剖分的截断或6-正则图的弧传递二面体方案截断得到。
Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvorak, Kral and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this res…
Short rainbow cycles for families of small edge sets
本文扩展相关结果，证明小比例非星型颜色类即可保证图的对数级彩虹围长，还确定了彩虹围长从线性过渡到对数级的颜色类类型阈值比例。
We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are criti…
By Hall's marriage theorem, we study lower bounds of the Lin-Lu-Yau curvature of amply regular graphs with girth $3$ or $4$ under different parameter restrictions. As a consequence, we show that each conference graph has positive Lin-Lu-Yau curvature. Our approach also provides a geometric proof of a classical diameter estimate for amply regular graphs in the case of girth $4$ and some special cases of girth $3$.
Ricci-flat cubic graphs with girth five
本文分类了所有围长至少为5的连通简单3-正则Ricci平坦图，证明仅有Petersen图、Triplex和十二面体图，并更正了Lin-Lu-Yau在2014年遗漏Triplex的分类。
We construct an infinite family of (q+1)-regular Ramanujan graphs X_n of girth 1. We also give covering maps X_{n+1} --> X_n such that the minimal common covering of all the graphs is the universal covering tree.
We prove that every 3-regular graph with no circuit of length less than six has a subgraph isomorphic to a subdivision of the Petersen graph.
New upper bounds on the order of mixed cages of girth 6
本文研究围长为6的混合笼的阶，提出了一族无限混合图，在部分情况下改进了已有结果的上界。
We study the Cage Problem for regular and biregular planar graphs. A $(k,g)$-graph is a $k$-regular graph with girth $g$. A $(k,g)$-cage is a $(k,g)$-graph of minimum order. It is not difficult to conclude that the regular planar cages are the Platonic Solids. A $(\{r,m\};g)$-graph is a graph of girth $g$ whose vertices have degrees $r$ and $m.$ A $(\{r,m\};g)$-cage is a $(\{r,m\};g)$-graph of minimum order. In this case we determine the triplets of values $(\{r,m\};g)$ for which there exist planar $(\{r,m\};g)$--graphs, for all those values we construct examples. Furthermore, for many triplet…
Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the the number of edges of $C\!D(k,q)$, then $e =Ω(v^{1+ {1\over {k-t+1}}})$. These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order $v$ and girth at least $g$, $ g\ge 5$, $g \not= 11,12$. For $g\ge 24$, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for $5\le …
Cubic vertex-transitive graphs of girth seven
本文通过图的签名对围长为7的三次点传递图进行完整分类，并证明若每条边包含在相同数量的7-圈中，则该图也是弧传递的。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。