Hany Ibrahim, Peter Tittmann
A graph $G$ is $H$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to $H$. Given a graph $H$, we present sufficient and necessary conditions for a graph $G$ such that $G/e$ is $H$-free for any edge $e$ in $E(G)$. Thereafter, we use these conditions to characterize forests, claw-free, $2K_{2}$-free, $C_{4}$-free, $C_{5}$-free, split, and pseudo-split graphs.
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For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either forced or forbidden edges. Among the things we show, we determine the structure of the forced or forbidden edge sets, the relationship between the sizes of forced and forbidden sets for a sequence, and the resulting structural consequences to their realizations. This includes showing that the diameter of every realization of a degree sequence containing forced…
It has been conjectured that whenever an optimal covering array exists there is also a uniform covering array with the same parameters and this is true for all known optimal covering arrays. When used as a test suite, the application context may have pairs of parameters that must be avoided and Covering arrays avoiding forbidden edges (CAFE) are a generalization accommodating this requirement. We prove that there is an arc-transitive, highly symmetric constraint graph where the unique optimal covering array avoiding forbidden edges is not uniform. This does not refute the conjecture but it doe…
The Graph Minor Theorem of Robertson and Seymour asserts that any graph property, whatsoever, is determined by an associated finite list of graphs. We view this as an impressive generalization of Kuratowski's theorem, which characterizes planarity in terms of two forbidden subgraphs, $K_5$ and $K_{3,3}$. Robertson and Seymour's result empowers students to devise their own Kuratowski type theorems; we propose several undergraduate research projects with that goal. As an explicit example, we determine the seven forbidden minors for a property we call strongly almost--planar (SAP). A graph is SAP…
Let $\mathcal{H}$ be a class of given graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no induced copies of $H$ for any $H \in \mathcal{H}$. In this article, we characterize all pairs $\{R,S\}$ of graphs such that every connected $\{R,S\}$-free graph has the same edge-connectivity and minimum degree.
A forbidden pair for quasi 5-contractible edges
本文定义了准$k$连通图中准$k$可收缩边的禁用对概念，并证明$\{K_{4}^{-}, \overline{P_{5}}\}$是准5可收缩边的一个禁用对。
Structural Bounds and Forbidden Induced Subgraphs for Edge-Add Graph Classes
研究边添加图类的结构界限与禁止诱导子图，证明对于p-边添加弦图，禁止诱导子图集有限，而完美图不成立，并给出基类的结构界限。
Characterizing the forbidden pairs for graphs to be super-edge-connected
本文刻画了当禁止子图集合大小为1或2时，使得所有不含这些诱导子图的图均为超边连通的所有禁止子图集合（除特殊图外）。
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the tropical intersections problem derived from the Kuramoto equations. Regular subdivisions of adjacency polytopes are instrumental in solving these problems. This paper explores connections between the regular subdivisions of an adjacency polytope and the contraction of the underlying graph along an edge. We construct a special …
Forbidden subdivision in integral trees
该研究针对整树的结构性质，证明了所有特征值为整数的树不存在含7个顶点的细分边，为整树的结构刻画提供了新的禁用子结构结论。
Nowhere-zero $3$-flows in graphs with forbidden edge-cuts
本文研究Tutte的3-流猜想，在允许5-边割时，证明无6至40大小边割的4-边连通图存在无处为零3-流，并推广到本质41-边连通情形。
In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$, and $k$, where $n-1 \leq k \leq \binom{n}{2}$, what is the minimum possible number of vertices in a $d$-regular undirected graph whose subgraphs with $n$ vertices contain at most $k$ edges? The goal of this paper is to give the exact number of vertices for each instance of the problem and also to provide some bounds for general values of $n$, $d$, and $k$. …
A cycle is a graph is dominating if every edge of the graph is incident with a vertex of the cycle. In this paper, we investigate the characterization of the class of the forbidden pairs guaranteeing the existence of a dominating cycle and show the following two results: (i) Every $2$-connected $\{P_{5}, K_{4}^{-}\}$-free graph contains a longest cycle which is a dominating cycle. (ii) Every $2$-connected $\{P_{5}, W^{*}\}$-free graph contains a longest cycle which is a dominating cycle. Here $P_{5}$ is the path of order $5$, $K_{4}^{-}$ is the graph obtained from the complete graph of order $…
Given a family of graphs $\mathcal{H}$, a graph $G$ is $\mathcal{H}$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to any graph in $\mathcal{H}$. We present sufficient and necessary conditions for a graph $G$ such that $G/e$ is $\mathcal{H}$-free for any edge $e$ in $E(G)$. Thereafter, we use these conditions to characterize claw-free and line graphs.
A cycle in a graph is called dominating if every edge of the graph is incident with a vertex of the cycle. In this paper, we investigate forbidden pairs guaranteeing the existence of a dominating cycle in 2-connected graphs.
A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end vertices and deleting all edges between every two vertex except one edge to gain again a simple graph and is denoted bye G/e. In this paper we prove some theorems related to a graph and its contraction. For example we prove the following theorem. If G is a connected graph that has a spanning k-end tree and |V (G)| > K + 1 then there exist an edge e such G/e …
Maximizing the number of stars in graphs with forbidden properties
本文将非Hamilton图中边数与t-星形数量的极值结果推广至可迹性、Hamilton连通性等性质，确定了各禁用性质的极图族，并找到了非k-连通图中t-星形的最大数量。
For a set of five edges, a graph splits if one of the associated Dodgson polynomials is equal to zero. A graph G splitting for every set of five edges is a minor-closed property. As such there is a finite set of forbidden minors F such that if a graph H does not contain a minor isomorphic to any graph in F, then H splits. In this paper we prove that if a graph G is simple, 3-connected, and splits, then G must not contain any minors isomorphic to K5, K3,3, the octahedron, the cube, or a graph that is a single delta-Y transformation away from the cube. As such this is the set of all simple 3-con…
On the difference of mean subtree orders under edge contraction
本文研究树边收缩对平均子树阶的影响，完整证明了 Jamison 于 1983 年提出的猜想：任意边收缩使平均子树阶至少减少 1/3，从而将已有悬挂边情形推广到一般情形。
We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental.
Chordal signed graphs and signed bigraphs
本文研究了弦符号图和弦符号双图的特征化性质，揭示了其与严格弦有向图的关系，并给出了禁止子图的特征化方法。
For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by $\varepsilon(G)$, is the sum of the eccentricity of individual vertices. In \cite{Kc}, the authors posed a conjecture which states that given a graph $G$ with at least three vertices, the difference between $W(G)$ and $\varepsilon(G)$ decreases when an edge is contracted and proved that the conjecture is true when $e$ is a bridge. In this manuscript, we confirm that the conjecture is true for any connected graph $G$ with at least three ve…
The line graph $Γ$ of a multi-graph $Δ$ is the graph whose vertices are the edges of $Δ$, where two such edges are adjacent if and only if they meet in a single vertex of $Δ$. We provide several characterizations of such line graphs and in particular show that a graph is a line graph if and only if it does not contain one of $33$ graphs, all of which correspond to bases of anisotropic vectors of a $6$-dimensional orthogonal geometry of $-$-type over a field with two elements, or, equivalently, to sets of $6$ generating reflections in the Weyl group of type $E_6$.
Contractions in perfect graph
本文刻画收缩任意边集后仍完美的图类，证明只需图本身完美且收缩任意单边后保持完美，并给出禁诱导子图刻画、多项式识别算法及等价的 utter graph 完美性判据。
In this paper, we consider the problem of reducing the semitotal domination number of a given graph by contracting $k$ edges, for some fixed $k \geq 1$. We show that this can always be done with at most 3 edge contractions and further characterise those graphs requiring 1, 2 or 3 edge contractions, respectively, to decrease their semitotal domination number. We then study the complexity of the problem for $k=1$ and obtain in particular a complete complexity dichotomy for monogenic classes.
We show that the norm graph constructed in [J. Kollár, L. Rónyai and T. Szabó, Norm-graphs and bipartite Turán numbers, Combinatorica, 16 (1996) 399--406] with $n$ vertices about $\frac{1}{2}n^{2-1/t}$ edges, which contains no copy of $K_{t,(t-1)!+1}$, does not contain a copy of $K_{t+1,(t-1)!-1}$.
Subgraphs in random graphs with specified degrees and forbidden edges
该研究通过更精确的边切换论证，改进了带指定度数和禁止边的简单随机图及二部随机图中，子图包含与边禁止条件概率的界，扩展了适用的度序列范围和已有研究结果。
Fix $k \ge 2$ and let $H$ be a graph with $χ(H) = k+1$ containing a critical edge. We show that for sufficiently large $n$, the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian.
Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $\mathbf{k}$ and $n$, there is a complete …
For a given graph $\mathcal{G}$ of order $n$ with $m$ edges, and a real symmetric matrix associated to the graph, $M\left(\mathcal{G}\right)\in\mathbb{R}^{n\times n}$, the interlacing graph reduction problem is to find a graph $\mathcal{G}_{r}$ of order $r<n$ such that the eigenvalues of $M\left(\mathcal{G}_{r}\right)$ interlace the eigenvalues of $M\left(\mathcal{G}\right)$. Graph contractions over partitions of the vertices are widely used as a combinatorial graph reduction tool. In this study, we derive a graph reduction interlacing theorem based on subspace mappings and the minmax theory. …
The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if $C$ is a hamiltonian cycle of a graph $G$, then the square of $C$ is called a hamiltonian square of $G$. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square, in a 4-connected graph. The connectivity condition is necessary as, except $K_3$ and $K_4$, the square of a cycle is always 4-connected.
In this note, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by using only one edge contraction? We show that the problem is polynomial-time solvable on $P_3+kP_2$-free graphs for any $k \geq 0$ which combined with results of [1,2] leads to a complexity dichotomy of the problem on $H$-free graphs.
Hardness and Structural Properties of Fuzzy Edge Contraction
本文证明模糊图中的边删除和边收缩问题在多种阈值语义及受限图类下均为NP困难，并揭示平面性、串并联性等经典性质在模糊情形下的计算难解性。
In this paper, we study the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by at least one using $k$ edge contractions, for some fixed integer $k \geq 0$? We present positive and negative results regarding the computational complexity of this problem.
For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsym…
In this paper we deal with a Turán-type problem: given a positive integer n and a forbidden graph H, how many edges can there be in a graph on n vertices without a subgraph H? How does a graph look like if it has this extremal edge number? The forbidden graph in this article is a clique-path: a path of length k where each edge is extended to an r-clique, r >2. We determine both the extremal number and the extremal graphs for sufficiently large n.
For every graph $H$, there exists a polynomial-time algorithm deciding if a planar input graph $G$ can be contracted to~$H$. However, the degree of the polynomial depends on the size of $H$. In this paper, we identify a class of graphs $\cal C$ such that for every $H \in \cal C$, there exists an algorithm deciding in time $f(|V(H)|) \cdot |V(G)|^{\bigO{1}}$ whether a planar graph $G$ can be contracted to~$H$. (The function $f(\cdot)$ does not depend on $G$.) The class $\cal C$ is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph $H \in \cal …
Maximizing the number of cliques in $K_{r+1}$-free graphs with forbidden properties
本文研究$K_{r+1}$-自由图中避免若干哈密顿相关性质时的最大边数和$t$-团数，推广了先前结果，并确定了所有极值图。
Given a graph $G$, the strong clique number $ω_2'(G)$ of $G$ is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in $G$. We study the strong clique number of graphs missing some set of cycle lengths. For a graph $G$ of large enough maximum degree $Δ$, we show among other results the following: $ω_2'(G)\le5Δ^2/4$ if $G$ is triangle-free; $ω_2'(G)\le3(Δ-1)$ if $G$ is $C_4$-free; $ω_2'(G)\leΔ^2$ if $G$ is $C_{2k+1}$-free for some $k\ge 2$. These bounds are attained by natural extremal examples. Our work extends and improves upon previous wo…
We consider the following problem: for a given graph G and two integers k and d, can we apply a fixed graph operation at most k times in order to reduce a given graph parameter $π$ by at least d? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed d=1 and when restricted to chordal graphs. We also give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and d is fixed. Further, we complete the complexity dich…
We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.
Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed color patterns ensures larger monochromatic cliques. Specifically, it claims that for any fixed integer k and any clique K on k vertices edge-colored with two colors, there is a positive constant a such that in any complete n-vertex graph edge-colored with two colors that does not contain a copy of K, there is a monochromatic cli…
The majority of a cactus is prickly
该研究通过禁止诱导子图刻画了具有保边近一致运算的仙人掌图及森林，证明其存在对称保边多数运算并完成显式构造，为图论相关研究提供了新的刻画方法。
We start by building up some theory to state Wagner's Theorem, and then prove it using Kuratowski's Theorem, a proof of which is found in Diester (2000). Following this, we establish some connections between the chromatic number of a graph and some of its forbidden minors. The idea is that if we forbid $G$ to have certain graphs as a minor, then the chromatic number of $G$ cannot be too large. Intuitively, this makes sense: if we disallow $G$ from having too many edges, then this makes it easier to colour the graph with fewer colours; we will of course make this precise. We close by explaining…
The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ε>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ε$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every …
We give a characterization of 3-connected graphs which are planar and forbid cube, octahedron, and $H$ minors, where $H$ is the graph which is one $Δ-Y$ away from each of the cube and the octahedron. Next we say a graph is Feynman 5-split if no choice of edge ordering gives an obstruction to parametric Feynman integration at the fifth step. The 3-connected Feynman 5-split graphs turn out to be precisely those characterized above. Finally we derive the full list of forbidden minors for Feynman 5-split graphs of any connectivity.
A tree in an edge-colored connected graph $G$ is called \emph{a rainbow tree} if no two edges of it are assigned the same color. For a vertex subset $S\subseteq V(G)$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$. A \emph{$k$-rainbow coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a rainbow $S$-tree in $G$. The minimum number of colors that are needed in a $k$-rainbow coloring of $G$ is the \emph{$k$-rainbow index} of $G$, denoted by $rx_k(G)$. The \emph{Steiner distance $d(S)$} of a set $S$ of vertice…
On graphs which have locally complete 2-edge-colourings and their relationship to proper circular-arc graphs
该研究聚焦具有局部完全2边着色的图，证明其可多项式时间识别，给出类似Gallai余可比图刻画的禁用子结构特征，并刻画了对应类别的正常区间图与正常圆弧图。
The weighted $\mathcal{T}$-free $2$-matching problem is the following problem: given an undirected graph $G$, a weight function on its edge set, and a set $\mathcal{T}$ of triangles in $G$, find a maximum weight $2$-matching containing no triangle in $\mathcal{T}$. When $\mathcal{T}$ is the set of all triangles in $G$, this problem is known as the weighted triangle-free $2$-matching problem, which is a long-standing open problem. A main contribution of this paper is to give a first polynomial-time algorithm for the weighted $\mathcal{T}$-free $2$-matching problem under the assumption that $\ma…
We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link condition preserves homeomorphism in low dimensional complexes, and homotopy in general. But, checking the link condition involves computation in all dimensions, and hence can be costly, especially in high dimensional complexes. We define a weaker and more local condition called the p-link condition for each dimension p, and study its effect on edge contra…
In this paper, we consider the following problem: given a connected graph $G$, can we reduce the domination number of $G$ by one by using only one edge contraction? We show that the problem is $\mathsf{NP}$-hard when restricted to $\{P_6,P_4+P_2\}$-free graphs and that it is $\mathsf{coNP}$-hard when restricted to subcubic claw-free graphs and $2P_3$-free graphs. As a consequence, we are able to establish a complexity dichotomy for the problem on $H$-free graphs when $H$ is connected.
Further Results on the Maximum Number of Stars in Graphs with Forbidden Properties
本文研究不含k-边哈密顿性质图中t-星的最大数量，验证并改进了Berikkyzy等人的猜想，给出了小t值下的极图刻画。
A result due to Gyárfás, Hubenko, and Solymosi (answering a question of Erdös) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a complete subgraph on at least $\frac{c^2}{10}n$ vertices. In this paper we suggest a "higher-dimensional" analogue of the notion of an induced $K_{2,2}$ which allows us to generalize their result to $k$-uniform hypergraphs. Our result also has an interesting consequence in discrete geometry. In particular, it implies that the fractional Helly theorem can be derived as a …
Let $ρ(G)$ be the spectral radius of a graph $G$ with $m$ edges. Let $S_{m-k+1}^{k}$ be the graph obtained from $K_{1,m-k}$ by adding $k$ disjoint edges within its independent set. Nosal's theorem states that if $ρ(G)>\sqrt{m}$, then $G$ contains a triangle. Zhai and Shu showed that any non-bipartite graph $G$ with $m\geq26$ and $ρ(G)\geqρ(S_{m}^{1})>\sqrt{m-1}$ contains a quadrilateral unless $G\cong S_{m}^{1}$ [M.Q. Zhai, J.L. Shu, Discrete Math. 345 (2022) 112630]. Wang proved that if $ρ(G)\geq\sqrt{m-1}$ for a graph $G$ with size $m\geq27$, then $G$ contains a quadrilateral unless $G$ is o…
The core vertex set of a graph is an invariant of the graph. It consists of those vertices associated with the non-zero entries of the nullspace vectors of a $\{0,1\}$-adjacency matrix. The remaining vertices of the graph form the core--forbidden vertex set. For graphs with independent core vertices, such as bipartite minimal configurations and trees, the nullspace induces a well defined three part vertex partition. The parts of this partition are the core vertex set, their neighbours and the remote core--forbidden vertices. The set of the remote core--forbidden vertices are those not adjacent…
A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to rev…
Single-conflict colorings of degenerate graphs
退化图的单冲突着色问题通过证明在特定条件下，使用$O(\sqrt{d} \log n)$种颜色即可避免所有禁止颜色对。
We introduce a characterization for split graphs by using edge contraction. Then, we use it to prove that any ($2K_{2}$, claw)-free graph with $α(G) \geq 3$ is a split graph. Also, we apply it to characterize any pseudo-split graph. Finally, by using edge contraction again, we characterize unbalanced split graphs which we use to characterize the Nordhaus-Gaddum graphs.
The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. In this paper, we study the cop number in the classes of graphs defined by forbidding one or more graphs as either subgraphs or induced subgraphs. In the case of a single forbidden graph we completely characterize (for bo…
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. We examine the effects on $D(G)$ and $D'(G)$ when $G$ is modified by operations on vertex and edge of $G$. Let $G$ be a connected graph of order $n\geq 3$. We show that $-1\leq D(G-v)-D(G)\leq D(G)$, where $G-v$ denotes the graph obtained from $G$ by removal of a vertex $v$ and all edges incident to $v$ and these inequalities are true for the distinguishing index. Also we prove that $|D(…
Let $G$ be a graph of order $n$ and let $q\left( G\right) $ be the largest eigenvalue of the signless Laplacian of $G$. Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an independent set of order $n-k;$ and let $S_{n,k}^{+}$ be the graph obtained by adding an edge to $S_{n,k}.$ It is shown that if $k\geq2,$ $n\geq400k^{2},$ and $G$ is a graph of order $n,$ with no cycle of length $2k+2,$ then $q\left( G\right) <q\left( S_{n,k}^{+}\right) ,$ unless $G=S_{n,k}^{+}.$ This result completes the proof of a conjecture of de Freitas, Nikifo…
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