Yair Glasner
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.
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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.
Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether high-girth Ramanujan graphs have improved vertex expansion. Our results are two-fold: 1. For every $d = p+1$ for prime $p$ and infinitely many $n$, we…
We show that for every prime $d$ and $α\in (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2α\log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by $(3/\sqrt{2})\sqrt{d}$, and many eigenvectors fully localized on small sets of size $O(|V|^α)$. This strengthens the results of Ganguly-Srivastava, who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the "scarring" phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof ar…
We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window.
We prove that in sparse Erdős-Rényi graphs of average degree $d$, the vector chromatic number (the relaxation of chromatic number coming from the Lovàsz theta function) is typically $\tfrac{1}{2}\sqrt{d} + o_d(1)$. This fits with a long-standing conjecture that various refutation and hypothesis-testing problems concerning $k$-colorings of sparse Erdős-Rényi graphs become computationally intractable below the `Kesten-Stigum threshold' $d_{KS,k} = (k-1)^2$. Along the way, we use the celebrated Ihara-Bass identity and a carefully constructed non-backtracking random walk to prove two deterministic…
Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have 'high girth' and high chromatic number, generalizing a well known result about Ramanujan graphs.
We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and minimum degree $δ$ is at least $\tfrac{1}{g}(δ- 1)^{\lfloor \frac{g-1}{4}\rfloor}$. We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent $\tfrac{1}{4}g$ in this lower bound cannot be improved to $(\tfrac{1}{4}+\varepsilon)g$, we are also able to prove that it cannot be increased beyond $\frac{3}{8}…
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-…
Long-range expanders: construction and cutoff
本文证明每个Ramanujan图具有长程扩张，构造对数围长的显式长程扩张图，并确立固定度长程扩张图序列上随机游走的截断现象及其高斯极限轮廓，为传递谱扩张图的截断猜想提供中间进展。
We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs. Using this method we can show two results involving high girth spectral expander graphs. First, we show that given $d \geq 3$ and $n$, there exists an explicit distribution of $d$-regular $Θ(n)$-vertex graphs whe…
Very sharp distance and range transitions for random walk bridges on Ramanujan graphs
本文证明在具有对数围长的顶点传递 Ramanujan 图上，长度为对数阶的随机游走桥的最大距离在临界窗口内从根号对数阶突变至对数阶，并给出范围的两点极限定律。
Let $G$ be a large-girth $d$-regular graph and $μ$ be a random process on the vertices of $G$ produced by a randomized local algorithm. We prove the upper bound $(k+1-2k/d)\Bigl(\frac{1}{\sqrt{d-1}}\Bigr)^k$ for the (absolute value of the) correlation of values on pairs of vertices of distance $k$ and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the $d$-regular tree. In that case we give an explicit description for the (closure) of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infin…
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。
We prove that a random $d$-regular graph, with high probability, is a cut sparsifier of the clique with approximation error at most $\left(2\sqrt{\frac 2 π} + o_{n,d}(1)\right)/\sqrt d$, where $2\sqrt{\frac 2 π} = 1.595\ldots$ and $o_{n,d}(1)$ denotes an error term that depends on $n$ and $d$ and goes to zero if we first take the limit $n\rightarrow \infty$ and then the limit $d \rightarrow \infty$. This is established by analyzing linear-size cuts using techniques of Jagannath and Sen derived from ideas in statistical physics, and analyzing small cuts via martingale inequalities. We also prov…
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…
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 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.
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.
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.
We give explicit estimates between the spectral radius and the densities of short cycles for finite d-regular graphs. This allows us to show that the essential girth of a finite d-regular Ramanujan graph G is at least c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Using Benjamini-Schramm convergence this leads to a rigidity result saying that if most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. Kesten showed that if a Cayley …
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…
A detour of a graph G is a longest path in G. The detour order of G is the number of vertices in a detour of G. A graph is said to be detour-saturated if the addition of any edge increases strictly the detour order. L.W. Beineke, J.E. Dunbar and M. Frick asked the following three questions in 2005. (1) What is the smallest order of a detour-saturated graph of girth 4? (2) Let Pr be the graph obtained from the Petersen graph by splitting one of its vertices into three leaves. Is Pr the smallest triangle-free detour-saturated graph? (3) Does there exist a detour-saturated graph with finite girth…
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.
In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $δ$ there is an orientation of diameter at most $11\frac{n}{δ+1}+9$. In 2016, Surmacs reduced this bound to $7\frac{n}{δ+1}.$ In this paper, we consider the girth of a graph $g$ and show that for any $\varepsilon>0$ there is a bound of the form $(2g+\varepsilon)\frac{n}{h(δ,g)}+O(1)$, where $h(δ,g)$ is a polynomial. Letting $g=3$ and $\varepsilon<1$ gives an inprovement on the result by Surmacs.
We prove a sharp upper bound on the number of shortest cycles contained inside any connected graph in terms of its number of vertices, girth, and maximal degree. Equality holds only for Moore graphs, which gives a new characterization of these graphs. In the case of regular graphs, our result improves an inequality of Teo and Koh. We also show that a subsequence of the Ramanujan graphs of Lubotzky-Phillips-Sarnak have super-linear kissing numbers.
The $g$-girth-thickness $θ(g,G)$ of a graph $G$ is the smallest number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we calculate the $4$-girth-thickness $θ(4,G)$ of the complete $m$-partite graph $G$ when each part has an even number of vertices.
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。
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。
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.
We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph theorem, $E$ is the disjoint union of a $2$-factor and a perfect matching $M$. We refer to the edges of $M$ as chords and classify the cycles in $G$ by their number of chords. We define $γ_k(n)$ to be the largest integer $g$ such that every cubic $n$-vertex graph with a given perfect matching $M$ has a cycle of length at most $g$ with at most $k$ chords. Here we de…
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。
Given a graph $H$, a graph $G$ is called $H$-critical if $G$ does not admit a homomorphism to $H$, but any proper subgraph of $G$ does. Observe that $K_{k-1}$-critical graphs are the standard $k$-(colour)-critical graphs. We consider questions of extremal nature previously studied for $k$-critical graphs and generalize them to $H$-critical graphs. After complete graphs, the next natural case to consider for $H$ is that of the odd-cycles. Thus, given integers $\ell$ and $k$, $\ell\geq k$, we ask: what is the smallest order of a $C_{2\ell +1}$-critical graph of odd-girth at least $2k+1$? Denotin…
An Infinite Family of 6_Regular B-Cayley Graphs from the Petersen Graph
本文通过将彼得森图复制n次并按n阶循环置换连接顶点，构造出无限族6正则图{G_n}，证明G_3和G_4为拉马努金图，并将前五项提交至House of Graphs数据库。
A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4-cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity five, and thus girth five. We show by computer search that the smallest members of this class are three graphs with 76 vertices.
On the sum of the largest and smallest eigenvalues of graphs with high odd girth
该研究通过逼近理论证明了具有高奇数圈长的图的特征值之和上界，给出了γ_k的紧上界至多项对数因子。
Digraphs of Large Girth and Dichromatic Number in Tournaments with Large Dichromatic Number
本文证明每个二色数足够大的竞赛图包含具有大有向围长和大二色数的子有向图，并探讨了用底层图围长替代的情形，该情形蕴含Erdős-Hajnal猜想且对特定竞赛图族成立。
The distinguishing chromatic number, $χ_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $φ$, of $G$, such that the only automorphism of $G$ that preserves all colors of $φ$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $χ_D(G)\le Δ+1$. We prove this conjecture.
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。
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。
The limit points of the top and bottom eigenvalues of regular graphs
本文确定了正则图序列最大和最小特征值极限点的完整区间，并推广到前k个特征值向量，同时构造了具有大围长和局部化特征向量的几乎拉马努金图，加强了现有结果。
Optimization in graphical small cancellation theory
该研究聚焦图小消去理论，简化Osajda的概率方法，在构造含扩张图族的Gromov怪物群时显著减少生成元数量，推进相关群构造的优化。
Graphs with Large Girth and Small Cop Number
本文研究大围长或短圈少的图的警察数，证明其界限并推广Frankl工作，进而验证Meyniel猜想及弱Meyniel猜想在稠密图上的成立。
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…
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…
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 introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints, and apply this method in various scenarios. We establish a formula that generates a general upper bound for the fractional chromatic number of triangle-free graphs of maximum degree~$Δ\ge 3$. This upper bound matches that deduced from the fractional version of Reed's bound for small values of~$Δ$, and improves it when~$Δ\ge 17$, transitioning smoothly to the best possible asymptotic regime, barring a breakthrough in Ramsey theory. Focusing on sm…
Bounds for the smallest $k$-chromatic graphs of given girth
本文研究围长至少为 $g$ 的 $k$-色图的最小阶数 $n_g(k)$，通过穷举搜索和计算机算法给出新的上下界，并首次提供 $k=4$、$g>5$ 的小阶数实例。
A catalog of a class of (3,g) graphs for even girth g is introduced in this paper. A (k,g) graph is a regular graph with degree k and girth g. This catalog of (3,g) graphs for even girth g satisfying 6 <= g <= 16, has the following properties. Firstly, this catalog contains the smallest known (3, g) graphs. An appropriate class of cubic graphs for this catalog has been identified, such that the (3,g) graph of minimum order within the class is also the smallest known (3,g) graph. Secondly, this catalog contains (3,g) graphs for more orders than other listings. Thirdly, the class of graphs have …
We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth $g$ and size $m$ has a feedback vertex set of size at most $\frac{4m}{3g}$, improving the trivial bound of $\frac{2m}{g}$. We also prove that every $2$-connected graph with maximum degree $3$ and order $n$ has a feedback vertex set of size at most $\frac{n+2}{3}$.
A question of Erdős asks if for every pair of positive integers $r$ and $k$, there exists a graph $H$ having $\textrm{girth}(H)=k$ and the property that every $r$-colouring of the edges of $H$ yields a monochromatic cycle $C_k$. The existence of such graphs was confirmed by the third author and Ruciński. We consider the related numerical problem of determining the smallest such graph with this property. We show that for integers $r$ and $k$, there exists a graph $H$ on $R^{10k^2} k^{15k^3}$ vertices (where $R = R(C_k;r)$ is the $r$-colour Ramsey number for the cycle $C_k$) having $\textrm{girt…
On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs
本文研究循环多面体对应超图的无冲突色数，给出若干偶数维的精确渐近界及一般偶数维上下界，并揭示其与 Erdős 围长猜想的联系。
We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, $δ\le Z(G)$ where $δ$ is the minimum degree, in the triangle-free case. In particular, we show that $2 δ- 2 \le Z(G)$ for graphs with girth of at least 5, and this can be further improved when $G$ has a small cut set. Using these results, we are able to prove the Graph Complement Conjecture on minimum rank for a large class of graphs. Lastly, we make a conjecture that the lower bound for $Z(G)$ increases as a function of the girth, $g$, and $δ$.
On the sum of the largest and smallest eigenvalues of odd-cycle free graphs
本文研究无奇数环图中最大与最小邻接特征值之和，将 Csikvári 对奇围长 5 的界推广到一般奇围长 $k$，并给出奇围长 7 时更紧的数值上界。
We present an interactive framework that, given a membership test for a graph class $\mathcal{G}$ and a number $k$, finds and tests unavoidable sets for the class of graphs in $\mathcal{G}$ of path-width at most $k$. We put special emphasis on the case that $\mathcal{G}$ is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient pruning techniques. Using this framework we determine all extremal girth values of cubic graphs of path-width $k$ for all $k \in \{3,\dots, 10\}…
The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect $0$ if and only if it is $3$-edge-colourable, this invariant can measure how much a cubic graph differs from a $3$-edge-colourable graph. Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a $2$-factor. We show that there exist cyclically $5$-edge-connected snarks (cu…
We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to be implementable in practice due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalizes the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction we prove a strong upper bound on average degrees in small induced s…
Improved bounds for the smallest 4-chromatic graph of girth six
本文将围长为6的最小4-色图的阶数界改进为29≤n₆(4)≤64，构造出64顶点的见证图并通过形式化验证，同时给出结构障碍结果。
We denote by $χ$ g (G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P. Ossona de Mendez, A. Raspaud and X. Zhu, Decomposing a graph into forests, J. Graph Theory Ser. B, 102(1):38-52, 2012] and, independantly, from Wang and Zhang, [Y. Wang and Q. Zhang. Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5. One can ask if this bound of …
On the structure of graphs with given odd girth and large algebraic connectivity
该研究针对给定奇围长的图，以代数连通度为约束条件，证明了三类图在特定代数连通度阈值下必为二分图，并确定了相关常数的渐近最优性。
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