We prove that every 6-connected graph of girth $\geq 6$ has a $K_6$-minor and thus settle the Jorgensen conjecture for graphs of girth $ \geq 6$. Relaxing the assumption on the girth, we prove that every 6-connected $n$-vertex graph of size $\geq 3 1/5 n-8$ and of girth $\geq 5$ contains a $K_6$-minor.
The Turán type numbers for graphs without 3-cycles and 4-cycles are determined for vertex numbers from 40 to 49 inclusive. Hence, now, 43 of the first 50 numbers of OEIS A006856 are known. Estimates for the remaining seven numbers are presented.
To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let $\mathcal{G}(g,c,n)$ be the set of graphs $G$ with girth $g(G)=g$, circumference $c(G)=c$, and $n$ vertices; and let $\mathcal{H}(g,c,m)$ be the set of graphs with girth $g$, circumference $c$, and $m$ edges. In this work, we study the four following extremal problems on graphs: $A(g,c,n)=\min\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $B(g,c,n)=\max\{δ(G)\,|\; G \in \mathcal{G}(g,c,n) \}$, $α(g,c,m)=\min\{δ(G)\,|\; \in \mathcal{H}(g,c,m) \}$…
We utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper we also define the girth of a link, discuss relations to other knot invariants, and the possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that t…
For every $r \in \mathbb{N}$, let $θ_r$ denote the graph with two vertices and $r$ parallel edges. The $θ_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $θ_r$. This notion generalizes the usual concept of girth which corresponds to the case $r=2$. In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213--225, 2003], Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of $θ_{r}$-girth and …
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 prove properties of extremal graphs of girth 5 and order 20 <=v <= 32. In each case we identify the possible minimum and maximum degrees, and in some cases prove the existence of (non-trivial) embedded stars. These proofs allow for tractable search for and identification of all non isomorphic cases.
A signed graph $(G, Σ)$ is a graph $G$ and a subset $Σ$ of its edges which corresponds to an assignment of signs to the edges: edges in $Σ$ are negative while edges not in $Σ$ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of $(G,Σ)$ to $(H,Π)$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipar…
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…
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\}…
We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every $K_4$-minor free graph of odd-girth $2k+1$ admits a homomorphism to the projective hypercube of dimension $2k$. This supports a conjecture of the third author which generalizes the four-color theorem and relates to several outstanding conjectures such as Se…
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
Let $Γ=(G,σ)$ be a signed graph and $A(G,σ)$ be its adjacency matrix. Denote by $gr(G)$ the girth of $G$, which is the length of the shortest cycle in $G$. Let $r(G,σ)$ be the rank of $(G,σ)$. In this paper, we will prove that $r(G,σ)\geq gr(G)-2$ for a signed graph $(G,σ)$. Moreover, we characterize all extremal graphs which satisfy the equalities $r(G,σ)=gr(G)-2$ and $r(G,σ)=gr(G)$.
The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth $g$ at least 4, the cyclic edge-connectivity is bounded above by $(Δ-2)g$ where $Δ$ is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a $d$-regular graph of girth $g\geq4$ is sufficiently small, then the cyclic edge-connectivity is $(d-2)g$, providing a spectral condition for when this upper bound on cy…
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 prove the extremal function for $K_9^=$ minors, where $K_9^=$ denotes the complete graph $K_9$ with two edges removed. In particular, we show that any graph with $n$ vertices and at least $6n - 20$ edges either contains a $K_9^=$ minor or is isomorphic to a graph obtained from disjoint copies of $K_8$ and $K_{2, 2, 2, 2, 2}$ by identifying cliques of size 5. We utilize computer assistance to prove one of our lemmas.
We prove that every graph with $n$ vertices and at least $5n-8$ edges contains the Petersen graph as a minor, and this bound is best possible. Moreover we characterise all Petersen-minor-free graphs with at least $5n-11$ edges. It follows that every graph containing no Petersen minor is 9-colourable and has vertex arboricity at most 5. These results are also best possible.
We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\b…
We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with no $K_p$-minor satisfies $|E(G)|\leq (p-2)|V(G)| - (p-2)^2$.
Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdière parameter, excluded minors, and the spectral radius, J. Combin. Theory Ser. A 166 (2019) 42--58] determined the maximum spectral radius and characterized the unique extremal graph for $K_r$-minor free graphs of sufficiently large order $n$, he also made great progress on $K_{s,t}$-minor free graphs and posed a conjecture: Let $2\leq s\leq t$ and $n-s+1=pt+q$, where $n$ is sufficiently large and $1\leq q\leq t.$ Then $K_{s-1}\nabla (pK_t\cup K_q)$ is the unique extremal graph w…
Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_α(G)$ of $A (G)$ and $D (G) $ by $$A_α(G)=αD(G)+(1-α)A(G)$$ for any real number $0\leqα\leq1$. The \emph{$α$-index} of $G$ is the largest eigenvalue of $A_α(G)$. In this paper, we determine the maximum $α$-index and characterize all extremal graphs for $K_r$ minor-free graphs, $K_{s,t}$ minor-free graphs, and star-forest-free graphs for any $0<α<1$ by unified eigenvector approach, respectively.
For a graph $H$, let $c(H)=\inf\{c\,:\,e(G)\geq c|G| \mbox{ implies } G\succ H\,\}$, where $G\succ H$ means that $H$ is a minor of $G$. We show that if $H$ has average degree $d$, then $$ c(H)\le (0.319\ldots+o_d(1))|H|\sqrt{\log d} $$ where $0.319\ldots$ is an explicitly defined constant. This bound matches a corresponding lower bound shown to hold for almost all such $H$ by Norin, Reed, Wood and the first author.
For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying conjectures of Reed and Wood, and Harvey and Wood. We derive the above result from a theorem which allows us to find two vertex disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.
A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is described by a given frame template over a prime field. Subject to the matroid minors hypothesis of Geelen, Gerards, and Whittle, we use our result to determine, for each projective or affine geometry $N$ over a prime field $\mathbb F$, a best-possible upper bound on the number of elements in a simple $\mathbb F$-representable matroid $M$ of sufficiently large r…
For a graph $G$ with $p$ vertices the closed convex cone $\mathbb{S}^p_{\succeq0}(G)$ consists of all real positive semidefinite $p\times p$ matrices with zeros in the off-diagonal entries corresponding to nonedges of $G$. The extremal rays of this cone and their associated ranks have applications to matrix completion problems, maximum likelihood estimation in Gaussian graphical models in statistics, and Gauss elimination for sparse matrices. For a graph $G$ without $K_5$ minors, we show that the normal vectors to the facets of the $(\pm1)$-cut polytope of $G$ specify the off-diagonal entries …
Let $c(H)$ be the smallest value for which $e(G)/|G|\geq c(H)$ implies $H$ is a minor of $G$. We show a new upper bound on $c(H)$, which improves previous bounds for graphs with a vertex partition where some pairs of parts have many more edges than others -- for instance a complete bipartite graph with a small number of edges placed inside one class. We also show a tight matching lower bound for almost all such graphs. We apply these results to show $c(K_{ft/\log t,t}) = (0.638\dotsc+o_{f}(1))t\sqrt{f}$, for $f = o(\log t) = ω(1)$.
For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function $ex(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of $ex(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].
The "extremal function" $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the "density" of a graph $G$ is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs $H$ of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant densit…
School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）；School of Mathematics, East China University of Science and Technology（华东理工大学数学系）；School of Mathematics and Finance, Chuzhou University（滁州学院数学与金融学院）
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 $k+4$. Its exact girth $g=g(D(k,q))$ was conjectured in 1995 to be $k+5$ for odd $k$ and $q\geq4$. This conjecture was shown to be valid in 2016 when $\frac{k+5}{2}|_p(q-1)$, where $p$ is the characteristic of $\mathbb{F}_q$ and $m|_pn$ means that $m$ divides $p^r n$ for some nonnegative integer $r$. In this paper, for…
Laplacian coefficients of unicyclic graphs with the number of leaves and girth
设$G$是$n$阶图，$\mathcal{L}(G,λ)=\sum_{k=0}^n (-1)^{k}c_{k}(G)λ^{n-k}$是其Laplacian矩阵的特征多项式.由Ilić和Ilić猜想[A. Ilić，M. Ilić，Laplacian coefficients of trees with given number of leaves or vertices of degree two，线性代数及其应用431（2009）2195-2202.]本文研究了所有叶数为$l$的$n$ -顶点单圈图的集合$\mathcal{U}_{n,l}$中所有Laplacian系数最小的极图，其中$\mathcal{U}_{n,l}^g$表示叶数为$l$且围长为$g$的$n$ -顶点单圈图的集合$(\mathcal{U}_{n,l}^g, \preceq)$中极小元的性质。这些结果被用来反驳他们的猜想。此外，还研究了$\mathcal{U}_{n,l}^g$中具有最小类拉普拉斯能量的图。
Let $F_s$ be the friendship graph obtained from $s$ triangles by sharing a common vertex. For fixed $s\ge 2$ and sufficiently large $n$, the $F_s$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)],and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)]. Recently, the spectral extremal problems was widely studied for graphs containing no $H$ as a minor. For instance, Tait [J. Combin. Theory Ser. A 166 (2019)], Zhai and Lin [J. Combin. Theory Ser. B 157 (…
The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while holding the other two fixed. Here we focus in {\em bipartite Moore graphs\/}, that is, bipartite graphs attaining the optimum order, fixed either the degree/diameter or degree/girth. The fact that there are very few bipartite Moore graphs suggests the relaxation of some of the constraints implied by the bipartite Moore bound. First we deal with {\em local bipartit…
A classical result by Erdős, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a major open problem in extremal graph theory to decide if this bound is tight for all $k$. In this paper, we study the effect of forbidding short even cycles in incidence graphs of point-line arrangements in the plane. It is not known if the Erdős upper bound stated above can be improved to $o(n^{1+1/k})$ in this geometric setting, and in this note, we establish no…
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…
This paper considers the degree-diameter problem for extremal and largest known undirected circulant graphs of degree 2 to 9 of arbitrary diameter. As these graphs are vertex transitive it is possible to define their distance partition. The number of vertices in each level of the distance partition is shown to be related to an established upper bound for the order of Abelian Cayley graphs. Furthermore these graphs are all found to have odd girth which is maximal for their diameter. Therefore the type of each vertex in a level may be well-defined by the number of adjacent vertices in the preced…
