Degree powers in graphs with forbidden even cycle
Vladimir Nikiforov
本文证明了Yuster和Caro关于无指定偶圈的n阶图的度的p-幂和的一个猜想。我们的证明是基于一个新的充分条件的长路径，这可能是有用的，以及在其他应用。
We prove a conjecture of Yuster and Caro about the sum of the p-powers of the degrees of a graph of order n without a specified even cycle. Our proof is based on a new sufficient condition for long paths, that may be useful in other applications as well.
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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.
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Let G be a graph of order n and let q(G) be that largest eigenvalue of the signless Laplacian of G. In this note it is shown that if k>1 and q(G)>=n+2k-2, then G contains cycles of length l whenever 2<l<2k+3. This bound is asymptotically tight. It implies an asymptotic solution to a recent conjecture about the maximum q(G) of a graph G with no cycle of a specified length.
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Let $q\left( G\right) $ be the $Q$-index (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 $n-k.$ The main result of this paper is the following theorem: Let $k\geq3,$ $n\geq110k^{2},$ and $G$ be a graph of order $n$. If $G$ has no $C_{2k+1},$ then $q\left( G\right) <q\left( S_{n,k}\right) ,$ unless $G=S_{n,k}.$ This result proves the odd case of the conjecture in [M.A.A. de Freitas, V. Nikiforov, and L. Patuzzi, Maxima of the $Q$-index: forbidden $…
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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…
Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$ contains no directed cycle of length $r$ modulo $k$, then $D$ can be vertex-colored with $k$ colors so that each color class induces an acyclic subdigraph in $D$. The first result gives an affirmative answer to a question posed by Tuza in 1992, and the second implies the following strong form of a conjecture of Diwan, Kenkre and Vishwanathan: If an undirected…
We call a graph $G$ pancyclic if it contains at least one cycle of every possible length $m$, for $3\le m\le |V(G)|$. In this paper, we define a new property called chorded pancyclicity. We explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains at least one chorded cycle of every possible length $4, 5, \ldots, |V(G)|$. In particular, certain paths and triangles with pendant paths are forbidden.
Degree powers in graphs with forbidden subgraphs
Yuster和Caro首先研究了具有禁止子图的图的度的幂和。我们解决了他们的两个问题。
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Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings
本文通过无符号拉普拉斯谱半径研究禁止4-环和星形嵌入的图结构，提出新的极值结果并完善了相关谱定理。
An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision
本研究针对k≥2的k-连通图的Smith最长圈猜想，结合拉姆齐理论细化与计算机搜索，将其边界从Ω(k^(2/3))改进至Ω(k^(8/11))。
Let $G=(V,E)$ be a graph of order $n$ and let $γ_{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $δ\geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $γ_{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k…
Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups
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On extremal values of some degree-based topological indices with a forbidden or a prescribed subgraph
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For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either forced or forbidden edges. Among the things we show, we determine the structure of the forced or forbidden edge sets, the relationship between the sizes of forced and forbidden sets for a sequence, and the resulting structural consequences to their realizations. This includes showing that the diameter of every realization of a degree sequence containing forced…
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A container theorem for general digraphs with forbidden subdigraphs
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Maxima of the index: forbidden unbalanced cycles
本文研究了不含特定长度不平衡圈的符号图的最大指数问题，证明了在阶为$n$的不平衡符号图中，当谱半径不小于特定图$Γ_1$时必含长度为$r$的不平衡圈，扩展了Turán谱问题并部分解决了Lin和Wang提出的开放问题。
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The Caccetta-Haggkvist conjecture made in 1978 asserts that every orgraph on n vertices without oriented cycles of length <= l must contain a vertex of outdegree at most (n-1)/l. It has a rather elaborate set of (conjectured) extremal configurations. In this paper we consider the case l=3 that received quite a significant attention in the literature. We identify three orgraphs on four vertices each that are missing as an induced subgraph in all known extremal examples and prove the Caccetta-Haggkvist conjecture for orgraphs missing as induced subgraphs any of these orgraphs, along with cycles …
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Forbidden paths and cycles in the undirected underlying graph of a 2-quasi best match graph
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本文刻画了在边数为 $m$ 且禁止 $C_6^{\triangle}$ 子图的图中谱半径最大的极图，并给出谱半径阈值条件，保证图包含所有长度 3 到 7 的圈。
Forbidden subgraphs generating a finite set of graphs with minimum degree three and large girth
本文针对图论中禁用子图与图族有限性的问题，刻画了直径≥7的非毛虫树T，使得不含C3、C4和T作为诱导子图、最小度≥3的连通图族是有限的。
Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2ω - 3, χ equals its maximum clique size. We also give examples to show that the condition is necessary.
In a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs. For this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. In this paper we focus on conditions involving claws, nets and bulls as induced subgraphs. We extend Hamiltonicity results for finite claw-free and net-free graphs by Shepherd to locally finite graphs. Moreover, we generalise a classification of finite claw-free and net-free gra…
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本文研究了无向2-近最佳匹配图的结构特性，证明其不含长路径或循环作为诱导子图，并探讨了其支配双 clique分解的性质。
Given a positive integer $p$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Turán-type problem for $e_p(G)$: Given a positive integer $p$ and a graph $H$, determine the function $ex_p(n,H)$, which is the maximum value of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Turán number. Caro and Yuster determined the function $ex_p(n, P_\ell)$ for sufficiently large $n$, where $p\geq 2$ and $P_\ell$ denotes the path…
Limits of degeneracy for colouring graphs with forbidden minors
本文受 Hadwiger 猜想启发，证明了在具有强次线性分隔符的单调图族中，足够大的有界最大度二分图 H 满足 Seymour 提出的关于禁止 minors 的退化度性质，且三个条件均不可省略。
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
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。
Forbidden cycles in metrically homogeneous graphs
本文明确描述了Cherlin度量齐次图目录中本原3-约束空间对应的禁用圈有限族，填补了先前研究未显式给出该圈族的空白，并探讨了其在半群值度量空间和齐次化中的潜在应用。
Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs
本文刻画了禁止特定小圈和θ图时，无悬挂点图中达到最大无符号拉普拉斯谱半径的极值图，并完整解决了模3三个同余类下的固定边数问题。
The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to label constraints in a group-labeled graph, which is a directed graph with each arc labeled by an element of a group. Recently, paths and cycles in group-labeled graphs have been investigated, such as packing non-zero paths and cycles, where "non-zero" means that the identity element is a unique forbidden label. In this paper, we present a solution to finding an $s$--$t$ path with two labels forbidden in a group-labeled g…
For a bipartite graph G, let h(G) be the largest t such that either G or the bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)= min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contain a cycle. By Forb(n, H) we denote a set of bipartite graphs with parts of sizes n each, that do not contain H as an induced bipartite subgraph respecting the sides. One can easily show that h(Forb(n,H))= O(n^{1-s}) for a positive s if H is not strongly acyclic. Here, we prove that h(Forb(n, H)) is linear in n for all stron…
For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$ is an induced subgraph of some of $P_6$, $K_1 \cup P_5$, $2P_3$, $Z_2$ or $K_1 \cup Z_1$. Furthermore, for $X$ chosen as $2K_1 \cup K_3$, we list all eight imperfect graphs belonging to the class; and for every other choice of $X$, we show that there are infinitely many such graphs. In addition, for $X$ chosen as $B_{1,2}$, we describe the structure of all …
Constructor--Blocker games forbidding even cycles
研究禁止偶圈的构造者-阻挡者博弈，通过有限域几何构造证明g(n,K_3,C_4)=Θ(n^{3/2})，并建立g(n,K_3,C_{2k})的上下界，揭示k=2与k≥3的结构差异，确定g(n,C_t,C_{2k})的精确阶。
We characterise the pairs of graphs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are perfect. Similarly, we characterise pairs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are $ω$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $ω$-colourability of all connected $\{ X, Y \}$-free graphs which are of independence at least $3$, distinct from an odd cycle, and of order at least $n_0$, and similar characterisations subject to ea…
We analyze the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the…
Random graphs with forbidden vertex degrees
本文研究了随机图G_{n,λ/n}，它的条件是所有顶点的度都在给定的非负整数的子集S中。在对S的某种假设下，顶点度的经验分布是渐近泊松分布，其中某个参数\mux作为S的某个“特征方程”的根给出，该方程使某个函数\psis（μ）最大化。在S上的一个假设下，我们得到了这样一个随机图的结构的部分描述，包括一个巨分支存在（或不存在）的条件。必要的假设在许多情况下是良性的，并应用于集合S的许多选择，包括（分别）偶数和奇数的集合。随机\emph{even}图与完全图K_n上的随机簇模型有关。
An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The classes [h,2,1] are closed by taking induced subgraphs, therefore each one can be characterized by a family of minimal forbidden induced subgraphs. In this paper we associate the minimal forbidden induced subgraphs for [h,2,1] which are VPT with (color) h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, eve…
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…
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