Degree powers in graphs with forbidden subgraphs

Forbidden Induced Subgraphs

In descending generality I survey: five partial orderings of graphs, the induced-subgraph ordering, and examples like perfect, threshold, and mock threshold graphs. The emphasis is on how the induced subgraph ordering differs from other popular orderings and leads to different basic questions.

Tight Chromatic Number for a class of Graphs with two forbidden subgraphs

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.

The number of graphs with large forbidden subgraphs

In this note, extending some results of Erdos, Frankl, Rodl, Alexeev, Bollobas and Thomason we determine asymptotically the number of graphs which do not contain certain large subgraphs. In particular, if H_1,...,H_n,... are graphs with chromatic numbers r_1,...,r_n,... and order o(log n), we dermine asymptotically the number of graphs of order n not containing H_n as a subgraph. We also give similar results for induced subgraphs.

Forbidden Subgraphs for Chorded Pancyclicity

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.

Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs II: paws

In this paper we extend a result about a sufficient condition for Hamiltonicity for finite graphs by Broersma and Veldmann to locally finite graphs. In order to do this we use topological circles within the Freudenthal compactification of a locally finite graph as infinite cycles. The condition we focus on in this paper is in terms of forbidden induced subgraphs, namely being claw-free and a relaxation of being paw-free.

Forbidden subgraphs and 2-factors in 3/2-tough graphs

A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ is a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free graph has a 2-factor. The toughness condition of this result is sharp. Moreover, for any $\varepsilon>0$ there exists a $(2-\varepsilon)$-tough $2P_5$-free graph without a 2-factor. This implies that the graph $P_4 \cup P_{10}$ is best possible for a forbidden subgraph in a sense.

Minimal separators in graph classes defined by small forbidden induced subgraphs

Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on set…

Hadwiger's Conjecture with Certain Forbidden Induced Subgraphs

We prove that $\{\overline{K_3}, H\}$-free graphs are not counterexamples to Hadwiger's Conjecture, where $H$ is any one of 33 graphs on seven, eight, or nine vertices, or $H=K_8$. This improves on past results of Plummer-Stiebitz-Toft, Kriesell, and Bosse. The proofs are mostly computer-assisted.

Difference of forbidden pairs containing a claw

When we study forbidden subgraph conditions guaranteeing graphs to have some properties, a claw (or $K_{1,3}$) frequently appears as one of forbidden subgraphs. Recently, Furuya and Tsuchiya compared two classes generated by different forbidden pairs containing a claw, and characterized one of such classes. In this paper, we give such characterization for three new classes. Furthermore, we give applications of our characterizations to some forbidden subgraph problems.

Disconnected Forbidden Subgraphs, Toughness and Hamilton Cycles

In 1974, Goodman and Hedetniemi proved that every 2-connected $(K_{1,3},K_{1,3}+e)$-free graph is hamiltonian. This result gave rise many other hamiltonicity conditions for various pairs and triples of forbidden connected subgraphs under additional connectivity conditions. In 1997, it was proved that a single forbidden connected subgraph $R$ in 2-connected graphs can create only a trivial class of hamiltonian graphs (complete graphs) with $R=P_3$. In this paper we prove that a single forbidden subgraph $R$ can create a non trivial class of hamiltonian graphs if $R$ is disconnected: $(\ast1)$ e…

Forbidden subgraphs in the norm graph

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 $K_{t+1,(t-1)!-1}$.

On coloring digraphs with forbidden induced subgraphs

We prove a conjecture by Aboulker, Charbit and Naserasr by showing that every oriented graph in which the out-neighborhood of every vertex induces a transitive tournament can be partitioned into two acyclic induced subdigraphs. We prove multiple extensions of this result to larger classes of digraphs defined by a finite list of forbidden induced subdigraphs. We thereby resolve several special cases of an extension of the famous Gyárfás-Sumner conjecture to directed graphs by Aboulker et al.

Maximising the Number of Cycles in Graphs with Forbidden Subgraphs

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.

On forbidden induced subgraphs for unit disk graphs

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…

Forbidden Induced Subgraphs for Bounded $p$-Intersection Number

A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if and only if $|S(u)\cap S(v)|\geq p$. We show that every minimal forbidden induced subgraph for the hereditary class ${\cal G}(d,p)$ of graphs whose $p$-intersection number is at most $d$, has order at most $3\cdot 2^{d+1}+1$, and that the exponential dependence on $d$ in this upper bound is necessary. For $p\in \{ d-1,d-2\}$, we provide more explicit results …

Universal graphs with a forbidden subgraph: Block path solidity

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.

On Validity of Reed Conjecture for Classes of Graphs with Two Forbidden Subgraphs

Reed Conjecture is open for more than 20 years now. Here we prove that Reed Conjecture is valid for (1) {P4UnionK1, Kite}-free graphs (2) {Chair, Kite}-free graphs (3) {K2UnionK2complement , H}-free graphs and (4) {2K2, M}-free graphs where H and M are graphs on six vertices each. Reed conjecture is still open in general.

Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: nets and bulls

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…

Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs

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…

On the zero forcing number of the complement of graphs with forbidden subgraphs

Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs.

常数支配数的禁止子图

Forbidden subgraphs for constant domination number

本文刻画了使所有连通无某子图族的图的支配数有常数上界的子图族特征。

图类的结构界限与禁止诱导子图对于边添加图类

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。

The Cops and Robber game on graphs with forbidden (induced) subgraphs

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…

Validity of Borodin and Kostochka Conjecture for classes of graphs without a single, forbidden subgraph on 5 vertices

Problem of finding an optimal upper bound for the chromatic no. of a graph is still open and very hard. Borodin and Kostochka Conjecture is still open and if proved will improve Brook bound on Chromatic no. of a graph. Here we prove Borodin & Kostochka Conjecture for (1) (P4 Union K1)-free (2) P5-free (3) Chair-free graphs and 4) graphs with dense neighbourhoods. Certain known results follow as Corollaries.

树独立数与禁止诱导子图：排除一个6顶点路径和一个(2,t)-双 clique

Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique

研究排除6顶点路径和(2,t)-双 clique的图的树独立数上限，推进Dallard等人猜想的进展。

关于具有禁止子图或指定子图的若干基于度的拓扑指标的极值

On extremal values of some degree-based topological indices with a forbidden or a prescribed subgraph

本文将 Xu 关于第二 Zagreb 指标在给定团数下的极值结果推广到其他基于度的拓扑指标，并借助子图计数加权和的方法研究禁止或指定子图条件下的极值问题。

弦图和弦双图的符号化

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。

Existence of $2$-Factors in Tough Graphs without Forbidden Subgraphs

For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that every $2K_2$-free $3/2$-tough graph with at least three vertices has a $2$-factor, and the toughness bound $3/2$ is best possible. In viewing $2K_2$, the disjoint union of two edges, as a linear forest, in this paper, for any linear forest $R$ on 5, 6, or 7 vertices, we find the sharp toughness bound $t$ such that every $t$-tough $R$-free graph on at least …

刻画2-连通图中生成 $\varTheta$-子图的禁用对

Characterizing forbidden pairs for spanning $\varTheta$-subgraphs of 2-connected graphs

本文刻画使所有2-连通 $\{R,S\}$-free 图均含生成 $\varTheta$-子图的连通图禁用对，并刻画相应极小反例。

Eigenvalues and forbidden subgraphs I

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.

Minimal forbidden induced subgraphs of graphs of bounded clique-width and bounded linear clique-width

In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.

Forbidden Subgraph Bounds for Parallel Repetition and the Density Hales-Jewett Theorem

We study a special kind of bounds (so called forbidden subgraph bounds, cf. Feige, Verbitsky '02) for parallel repetition of multi-prover games. First, we show that forbidden subgraph upper bounds for $r \ge 3$ provers imply the same bounds for the density Hales-Jewett theorem for alphabet of size $r$. As a consequence, this yields a new family of games with slow decrease in the parallel repetition value. Second, we introduce a new technique for proving exponential forbidden subgraph upper bounds and explore its power and limitations. In particular, we obtain exponential upper bounds for two-p…

关于含禁止子图的顶点Ramsey图

On vertex Ramsey graphs with forbidden subgraphs

本研究针对顶点Ramsey图的存在性问题，证明Nešetřil与Rödl提出的含禁止子图的顶点r-Ramsey图存在的充分条件，在颜色数r足够大时也是必要的，并将结果推广到图族及稠密二项随机图的典型存在性。

On the Caccetta-Haggkvist conjecture with forbidden subgraphs

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 …

Forbidden induced subgraphs for perfectness of claw-free graphs of independence number at least 4

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 …

Regular Graphs with Forbidden Subgraphs of $K_n$ with $k$ Edges

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 $k$. …

On Betti numbers of flag complexes with forbidden induced subgraphs

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…

美丽图的禁止诱导子图刻画

A forbidden-induced-subgraph characterization of beautiful graphs

本文证明美丽图等价于无诱导$C_4$、gem、net、watch或奇洞的图，即$C_4$-自由可比图，并给出基于闭邻域构造偏序的证明，带来多项式识别与精确障碍族。

Generalizing forbidden induced subgraph characterizations of high throttling numbers

Zero forcing is a process that models the spread of information throughout a graph as white vertices are forced to turn blue using a color change rule. The idea of throttling, introduced in 2013 by Butler and Young, is to optimize the trade-off between the number of initial blue vertices and the time taken to force all vertices to become blue. The original throttling number of a graph minimizes the sum of these two quantities and the product throttling number minimizes their product. In addition, weighted throttling changes the weights given to these two quantities when minimizing their sum. S…

Rainbow vertex-connection and forbidden subgraphs

A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\math…

素数阶元素图的禁止子图

Forbidden Subgraphs of Prime Order Element Graph

本文研究有限群上素数阶元素图的禁止子图特征，刻画了该图成为完美图、余图、弦图、无爪图和区间图的条件。

Characterization of forbidden subgraphs for bounded star chromatic number

The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbid…

生成有限个最小度为3且围长较大的图的禁用子图

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。

顶点着色与禁用导出子图的重构

Reconfiguration of vertex colouring and forbidden induced subgraphs

本文研究受禁用导出子图限制的k-可着色图在(k+1)-着色重构图中的连通性与直径，刻画了单个禁用导出子图情形下连通性的充要条件，并给出了两类双禁用导出子图图类的连通性结果。

Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs

A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.

含禁用子图的超图中的团数

The number of cliques in hypergraphs with forbidden subgraphs

本文研究禁用完全 r 部超图的 $(r-1)$-一致超图中 r 团的最大数量，借助超图移除引理证明上界，并用超图 Turán 数给出下界。

On forbidden induced subgraphs for K_{1,3}-free perfect graphs

Considering connected $K_{1,3}$-free graphs with independence number at least $3$, Chudnovsky and Seymour (2010) showed that every such graph, say $G$, is $2ω$-colourable where $ω$ denotes the clique number of $G$. We study $(K_{1,3}, Y)$-free graphs, and show that the following three statements are equivalent. (1) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is perfect. (2) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd cycle and which has independence number at least $3$ is $ω$-colourable…

以禁用子图刻画的共极大子群图

Co-maximal subgroup graph characterized by forbidden subgraphs

本文针对几类典型有限群的共极大子群图，推导其为多种禁用子图类的充要条件，完成有限幂零群情形的完整分类并给出对应分裂图的有限阿贝尔群结构，同时提出若干未解决问题。

Cops and Robbers on Graphs with a Set of Forbidden Induced Subgraphs

It is known that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a forest whose every component is a path. In this study, we characterize all sets $\mathscr{H}$ of graphs with some $k\in \mathbb{N}$ bounding the diameter of members of $\mathscr{H}$ from above, such that $\mathscr{H}$-free graphs, i.e. graphs with no member of $\mathscr{H}$ as an induced subgraph, are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs.…

圆图在分裂图中的禁用导出子图刻画

Forbidden induced subgraph characterization of circle graphs within split graphs

本文针对圆图在分裂图中的子类，首次给出了基于极小禁用导出子图的结构刻画，解决了该领域长期未决的图论问题。

Graphs with two trivial critical ideals

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, $Γ_{\leq k}$, are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for $Γ_{\leq 2}$, and we use this forbidden subgraphs to get a classification of the graphs in $Γ_{\leq 2}$. As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the end of this ar…

有限群增强幂图中的禁用子图

Forbidden subgraphs in enhanced power graphs of finite groups

本文研究有限群的增强幂图的禁用子图相关分类问题，完成了增强幂图为split、threshold图的有限群分类，以及对应弦图、cograph的幂零群分类并给出非幂零群实例，部分解答了Peter J. Cameron的问题。

Graph Isomorphism for Graph Classes Characterized by two Forbidden Induced Subgraphs

We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …

幂图的禁止子图进一步研究

Further study on forbidden subgraphs of power graph

本文进一步研究群的幂图的禁止子图，刻画了幂图为链图、无钻石图及两类特定自由图的有限群，并完全确定了幂图为{P5, P5补图}-自由的幂零群、直积群和有限单群。