Graphs Of Girth Research Articles

More about sparse halves in triangle-free graphs

One of Erdős’s conjectures states that every triangle-free graph on vertices has an induced subgraph on vertices with at most edges. We report several partial results towards this conjecture. In particular, we establish the new bound on the number of edges in the general case. We completely prove the conjecture for graphs of girth , for graphs with independence number and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph. Bibliography: 21 titles.

On the distance spectrum of minimal cages and associated distance biregular graphs

A (k,g)-cage is a k-regular simple graph of girth g with minimum possible number of vertices. In this paper, (k,g)-cages which are Moore graphs are referred as minimal (k,g)-cages. A simple connected graph is called distance regular (DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular (DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal (k,g)-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal (k,g)-cages we give a formula for distance spectral radius in terms of k and g, and also determine polynomials of degree ⌊g2⌋, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal (k,g)-cage of diameter d has d+1 distinct distance eigenvalues, and this partially answers a problem posed in [1]. We prove that every DBR graph is a 2-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal (k,g)-cages. Finally we determine the full distance spectrum of subdivision of some minimal (k,g)-cages.

Cubic vertex-transitive graphs of girth six

In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth 6 is obtained. It is proved that every such graph, with the exception of the Desargues graph on 20 vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a 6-regular graph with respect to an arc-transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than 6 are also discussed.

Density of C−4-critical signed graphs

A signed bipartite (simple) graph (G,σ) is said to be C−4-critical if it admits no homomorphism to C−4 (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study of C−4-critical signed graphs, we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to C−4. In particular, the 4-color theorem is equivalent to: Given a planar graph G, the signed bipartite graph obtained from G by replacing each edge with a negative path of length 2 maps to C−4.We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any C−4-critical signed graph on n vertices must have at least ⌈4n3⌉ edges. Moreover, we show that for each value of n≥9 there exists a C−4-critical signed graph on n vertices with either ⌈4n3⌉ or ⌈4n3⌉+1 many edges.As an application, we conclude that all signed bipartite planar graphs of negative girth at least 8 map to C−4. Furthermore, we show that there exists an example of a signed bipartite planar graph of girth 6 which does not map to C−4, showing 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena.

Ricci-flat Graphs with Girth Four

Lin-Lu-Yau introduced a notion of Ricci curvature for graphs and obtained a complete classification for all Ricci-flat graphs with girth at least five. In this paper, we characterize all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.

On the characterization of some algebraically defined bipartite graphs of girth eight

For any field F and polynomials f2,f3∈F[x,y], let ΓF(f2,f3) denote the bipartite graph with vertex partition P∪L, where P and L are two copies of F3, and (p1,p2,p3)∈P is adjacent to [l1,l2,l3]∈L if and only if p2+l2=f2(p1,l1) and p3+l3=f3(p1,l1). The graph Γ3(F)=ΓF(xy,xy2) is known to be of girth eight. When F=Fq is a finite field of odd characteristic or F=F∞ is an algebraically closed field of characteristic zero, the graph Γ3(F) is conjectured to be the unique one with girth at least eight among those ΓF(f2,f3) up to isomorphism. This conjecture has been confirmed for the case that both f2,f3 are monomials over Fq, and for the case that at least one of f2,f3 is a monomial over F∞. If one of f2,f3∈Fq[x,y] is a monomial, it has also been proved the existence of a positive integer M such that G=ΓFqM(f2,f3) is isomorphic to Γ3(FqM) provided G has girth at least eight. In this paper, these results are shown to be valid when the restriction on the polynomials f2,f3 is relaxed further to that one of them is the product of two univariate polynomials. Furthermore, all of such polynomials f2,f3 are characterized completely.

Circular coloring and fractional coloring in planar graphs

AbstractWe study the following Steinberg‐type problem on circular coloring: for an odd integer , what is the smallest number such that every planar graph of girth without cycles of length from to admits a homomorphism to the odd cycle (or equivalently, is circular ‐colorable). Known results and counterexamples on Steinberg's Conjecture indicate that . In this paper, we show that exists if and only if is an odd prime. Moreover, we prove that for any prime , We conjecture that , and observe that the truth of this conjecture implies Jaeger's conjecture that every planar graph of girth has a homomorphism to for any prime . Supporting this conjecture, we prove a related fractional coloring result that every planar graph of girth without cycles of length from to is fractional ‐colorable for any odd integer .

Sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate

A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k. It is known that every planar graph of girth 6 (equivalently, without 3-, 4-, and 5-cycles) is 2-degenerate. On the other hand, there exist planar graphs without 4 and 5-cycles such as a truncated dodecahedral graph that are not 2-degenerate. Furthermore, a truncated dodecahedral graph also contains none of 6-, 7-, 8-, and 9-cycles. This motivates us to find sufficient conditions for planar graphs without 4-cycles and 5-cycles to be 2-degenerate.In this work, we investigate the degeneracy of planar graphs without 4-, 5-, j-, and k-cycles where j∈{6,7,8,9} and k∈{10,11}. For each j∈{6,9}, we give an example of a 3-regular planar graph without 4-, 5-, j-, 10-, and 11-cycles. In contrast, we prove that every planar graph without 4-, 5-, j-, and k-cycles is 2-degenerate for each j∈{7,8} and k∈{10,11}.

Extremal Graphs on the Rupture Degree

For a given graph [Formula: see text], by [Formula: see text] and [Formula: see text] denote the order of the largest component and the number of connected components of [Formula: see text], respectively. The rupture degree of [Formula: see text] is an important combinatorial parameters, which defined as [Formula: see text]. Clearly, the smaller the value [Formula: see text] is, the better the connectivity of graph [Formula: see text] is. In this paper, the tree structures with minimum rupture degree be discussed, and the extremal graphs on the rupture degree in terms of graph girth are also characterized.

On the information ratio of graphs without high-degree neighbors

We consider the information ratio of graph based secret sharing schemes in a special case of graphs in which vertices of degree at least 3 are not connected by an edge. We prove that – after some trivial reduction of the original graph – the information ratio depends on the maximal value of the difference of the degree and the number of triangles containing a given vertex of the reduced graph. This result can be considered as a common generalization of previous results on the information ratio of special trees and graphs of girth at least 6.

Spectral Threshold for Extremal Cyclic Edge-Connectivity

The universal 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 universal cyclic edge-connectivity is bounded above by \((\Delta -2)g\) where \(\Delta \) is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a d-regular graph of girth \(g\ge 4\) is sufficiently small, then the universal cyclic edge-connectivity is \((d-2)g\), providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight.

Local classical MAX-CUT algorithm outperforms p=2 QAOA on high-girth regular graphs

The p-stage Quantum Approximate Optimization Algorithm (QAOAp) is a promising approach for combinatorial optimization on noisy intermediate-scale quantum (NISQ) devices, but its theoretical behavior is not well understood beyond p=1. We analyze QAOA2 for the maximum cut problem (MAX-CUT), deriving a graph-size-independent expression for the expected cut fraction on any D-regular graph of girth >5 (i.e. without triangles, squares, or pentagons).We show that for all degrees D≥2 and every D-regular graph G of girth >5, QAOA2 has a larger expected cut fraction than QAOA1 on G. However, we also show that there exists a 2-local randomized classical algorithm A such that A has a larger expected cut fraction than QAOA2 on all G. This supports our conjecture that for every constant p, there exists a local classical MAX-CUT algorithm that performs as well as QAOAp on all graphs.

Complexity and algorithms for injective edge-coloring in graphs

An injective k-edge-coloring of a graph G is an assignment of colors, i.e. integers in {1,…,k}, to the edges of G such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a k-coloring exists is called Injectivek-Edge-Coloring. We show that Injective 3-Edge-Coloring is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth 6. Injective 4-Edge-Coloring remains NP-complete for cubic graphs. For any k≥45, we show that Injectivek-Edge-Coloring remains NP-complete even for graphs of maximum degree at most 53k. In contrast with these negative results, we show that Injectivek-Edge-Coloring is linear-time solvable on graphs of bounded treewidth. Moreover, we show that all planar bipartite subcubic graphs of girth at least 16 are injectively 3-edge-colorable. In addition, any graph of maximum degree at most k/2 is injectively k-edge-colorable.

Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products

In this paper we study group actions on quasi-median graphs, or "CAT(0) prism complexes", generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a quasi-median graph X and define the contact graph \mathcal{C}X for these hyperplanes. We show that \mathcal{C}X is always quasi-isometric to a tree, generalising a result of Hagen [18], and that under certain conditions a group action G \curvearrowright X induces an acylindrical action G \curvearrowright \mathcal{C}X , giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto [5]. As an application, we exhibit an acylindrical action of a graph product on a quasi-tree, generalising results of Kim and Koberda for right-angled Artin groups [20, 21]. We show that for many graph products G , the action we exhibit is the "largest" acylindrical action of G on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth \geq 6 are equationally noetherian, generalising a result of Sela [27].

Emergence of the circle in a statistical model of random cubic graphs

We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3-regular graphs and making using of the Ollivier curvature, a coarse analogue of the Ricci curvature. Numerical analysis shows that the Hausdorff and spectral dimensions of the model approach 1 in the joint classical-thermodynamic limit and we argue that the scaling limit of the model is the circle of radius r, . Given mild kinematic constraints, these claims can be proven with full mathematical rigour: speaking precisely, it may be shown that for 3-regular graphs of girth at least 4, any sequence of action minimising configurations converges in the sense of Gromov–Hausdorff to . We also present strong evidence for the existence of a second-order phase transition through an analysis of finite size effects. This—essentially solvable—toy model of emergent one-dimensional geometry is meant as a controllable paradigm for the nonperturbative definition of random flat surfaces.

Regular graphs of girth 5 from elliptic semiplanes of type [formula omitted

A well-known technique to construct regular graphs with girth 5 is the amalgamation into the incidence graphs Cq and Lq, elliptic semiplanes of type C and L respectively, where q is a prime power. The case q odd has extensively been studied by means of amalgamations into Lq. In this paper we provide new families of small regular graphs of girth 5 constructed by amalgamation into Cq using finite fields of even order.

Distances in graphs of girth 6 and generalised cages

In this paper we present bounds on the radius and diameter of graphs of girth at least 6 and for (C4,C5)-free graphs, i.e., graphs not containing cycles of length 4 or 5. We show that the diameter of a graph G of girth at least 6 is at most 3nδ2−δ+1−1, and the radius is at most 3n2(δ2−δ+1)+10, where n is the order and δ the minimum degree of G. If δ−1 is a prime power, then both bounds are sharp apart from an additive constant.For graphs of large maximum degree Δ, we show that these bounds can be improved to 3n−Δδδ2−δ+1−3(δ−1)Δ(δ−2)δ2−δ+1+10 for the diameter, and 3n−3Δδ2(δ2−δ+1)−3(δ−1)Δ(δ−2)2(δ2−δ+1)+22 for the radius. We further show that only slightly weaker bounds hold for (C4,C5)-free graphs.As a by-product we obtain a result on a generalisation of cages. For given δ,Δ∈N with Δ≥δ let n(δ,Δ,g) be the minimum order of a graph of girth g, minimum degree δ and maximum degree Δ. Then n(δ,Δ,6)≥Δδ+(δ−1)Δ(δ−2)+32. If δ−1 is a prime power, then there exist infinitely many values of Δ such that, for δ constant and Δ large, n(δ,Δ,6)=δΔ+O(Δ).

Perfect Matching Index versus Circular Flow Number of a Cubic Graph

The perfect matching index of a cubic graph $G$, denoted by $\pi(G)$, is the smallest number of perfect matchings that cover all the edges of $G$. According to the Berge--Fulkerson conjecture, $\pi(G)\le5$ for every bridgeless cubic graph $G$. The class of graphs with $\pi\ge 5$ is of particular interest as many conjectures and open problems, including the famous cycle double cover conjecture, can be reduced to it. Although nontrivial examples of such graphs are very difficult to find, a few infinite families are known, all with circular flow number $\Phi_c(G)=5$. It has been therefore suggested [Abreu et al., Electron. J. Combin., 23 (2016), P3.54] that $\pi(G)\ge 5$ might imply $\Phi_c(G)\ge 5$. In this article we dispel these hopes and present a family of cyclically 4-edge-connected cubic graphs of girth at least 5 with $\pi\ge 5$ and $\Phi_c\le 4+\frac23$.

A Proof of Brouwer's Toughness Conjecture

The toughness $t(G)$ of a connected graph $G$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all proper subsets $S\subset V(G)$ such that $c(G-S)>1$, where $c(G-S)$ denotes the number of components of $G-S$. Let $\lambda$ denote the second largest absolute eigenvalue of the adjacency matrix of a graph. For any connected $d$-regular graph $G$, it has been shown by Alon that $t(G)>\frac{1}{3}(\frac{d^2}{d\lambda+\lambda^2}-1)$, through which, Alon was able to show that for every $t$ and $g$ there are $t$-tough graphs of girth strictly greater than $g$, and thus disproved in a strong sense a conjecture of Chv\'atal on pancyclicity. Brouwer independently discovered a better bound $t(G)>\frac{d}{\lambda}-2$ for any connected $d$-regular graph $G$, while he also conjectured that the lower bound can be improved to $t(G)\ge \frac{d}{\lambda} - 1$. We confirm this conjecture.

Choosability with separation of cycles and outerplanar graphs

We consider the following list coloring with separation problem of graphs: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|\le a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$ for any vertex $v$ and $\varphi(u)\cap \varphi(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth $g\ge 5$.