Free Graphs Research Articles

Cutting a tree with subgraph complementation is hard, except for some small trees

AbstractFor a graph property , Subgraph Complementation to is the problem to find whether there is a subset of vertices of the input graph such that modifying by complementing the subgraph induced by results in a graph satisfying the property . We prove that the problem of Subgraph Complementation to ‐free graphs is NP‐Complete, for being a tree, except for 41 trees of at most 13 vertices (a graph is ‐free if it does not contain any induced copies of ). This result, along with the four known polynomial‐time solvable cases (when is a path on at most four vertices), leaves behind 37 open cases. Further, we prove that these hard problems do not admit any subexponential‐time algorithms, assuming the Exponential‐Time Hypothesis. As an additional result, we obtain that Subgraph Complementation to paw‐free graphs can be solved in polynomial‐time.

Induced Subgraph Density. I. A loglog Step Towards Erd̋s–Hajnal

Abstract In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^{c}$, and they proved that this is true with $ |G|^{c}$ replaced by $2^{c\sqrt{\log |G|}}$. Until now, there has been no improvement on this result (for general $H$). We prove a strengthening: that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ with $|G|\ge 2$ has a clique or stable set of size at least $$ \begin{align*} &2^{c\sqrt{\log |G|\log\log|G|}}.\end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.

Turán problems in pseudorandom graphs

AbstractGiven a graph $F$ , we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$ . We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$ . Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$ -free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.

The optimal χ-bound for (P7,C4,C5)-free graphs

In this paper, we give an optimal χ-binding function for the class of (P7,C4,C5)-free graphs. We show that every (P7,C4,C5)-free graph G has χ(G)≤⌈119ω(G)⌉. To prove the result, we use a decomposition theorem obtained in Cameron et al. (2020) [4] combined with careful inductive arguments and a nontrivial use of the König theorem for bipartite matching.

Computational Study of the Effect of the Size-Dependent Dielectric Functions of Gold Nanomaterials on Optical Properties

The effect of size on the optical properties of gold nanomaterials has been studied using the theoretical Drude–Sommerfield model. The real and imaginary parts of the dielectric function of bulk as a function of wavelength due to free electron contribution and the real and imaginary parts of the dielectric function of nanogold materials as a function of wavelength due to free electron and bond electron contribution are calculated. The real and imaginary parts of the dielectric function of bulk as a function of wavelength due to the free electron contribution graph and The real and imaginary parts of the dielectric function of nanogold materials as a function of wavelength due to free electron and bond electron contributions are plotted. As we observed from the graphs, the real dielectric functions of both bulk and nanogold materials are inversely proportional to wavelength. The imaginary part of the dielectric function of bulk gold materials is independent of wavelength. At high wavelengths, the size of the gold nanomaterial is highly influenced by both real and imagined dielectric functions at high waves. As the wavelength increases, the effect of the size on the dielectric function also increases. The size-dependent dielectric function of nanomaterials is highly influenced by their optical properties and electrical structure.

Tree independence number I. (Even hole, diamond, pyramid)‐free graphs

AbstractThe tree‐independence number , first defined and studied by Dallard, Milanič, and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so‐called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass of (even hole, diamond, pyramid)‐free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that has bounded . Via existing results, this yields a polynomial‐time algorithm for the Maximum Weight Independent Set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič, and Štorgel that in a hereditary graph class, is bounded if and only if the treewidth is bounded by a function of the clique number.

The signless Laplacian spectral radius of graphs without intersecting odd cycles

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1,$ respectively, which intersect in exactly one common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

Contraction Decomposition in Unit Disk Graphs and Algorithmic Applications in Parameterized Complexity

We give a new decomposition theorem in unit disk graphs (UDGs) and demonstrate its applicability in the fields of Structural Graph Theory and Parameterized Complexity. First, our new decomposition theorem shows that the class of UDGs admits an “almost” Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010], and H -minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. Even bounded-degree UDGs can contain arbitrarily large cliques as minors, and therefore our result is a significant advance in the study of contraction decompositions. Additionally, this result answers an open question posed by Hajiaghayi ( www.youtube.com/watch?v=2Bq2gy1N01w ) regarding the existence of contraction decompositions for classes of graphs beyond H -minor free graphs though under a relaxation of the original formulation. Second, we present a “parameteric version” of our new decomposition theorem. We prove that there is an algorithm that, given a UDG G and a positive integer k , runs in polynomial time and outputs a collection of \(\mathcal {O}(k)\) tree decompositions of G with the following properties. Each bag in any of these tree decompositions can be partitioned into \(\mathcal {O}(k)\) connected pieces (we call this measure the chunkiness of the tree decomposition). Moreover, for any subset S of at most k edges in G , there is a tree decomposition in the collection such that S is well preserved in the decomposition in the following sense. For any bag in the tree decomposition and any edge in S with both endpoints in the bag, either its endpoints lie in different pieces or they lie in a piece that is a clique. Having this decomposition at hand, we show that the design of parameterized algorithms for some cut problems becomes elementary. In particular, our algorithmic applications include single-exponential (or slightly super-exponential) algorithms for well-studied problems such as Min Bisection , Steiner Cut , s -Way Cut , and Edge Multiway Cut-Uncut on UDGs; these algorithms are substantially faster than the best-known algorithms for these problems on general graphs.

Induced Subgraphs and Tree Decompositions VIII: Excluding a Forest in (Theta, Prism)-Free Graphs

Induced Subgraphs and Tree Decompositions VIII: Excluding a Forest in (Theta, Prism)-Free Graphs

Bipartite (P6,C6)-Free Graphs: Recognition and Optimization Problems

The canonical decomposition of a bipartite graph is a new decomposition method that involves three operators: parallel, series, and K⨁ S. The class of weak-bisplit graphs is the class of totally decomposable graphs with respect to these operators, and the class of bicographs is the class of totally decomposable graphs with respect to parallel and series operators. We prove in this paper that the class of bipartite (P6,C6)-free graphs is the class of bipartite graphs that are totally decomposable with respect to parallel and K⨁S operators. We present a linear time recognition algorithm for (P6,C6)-free graphs that is symmetrical to the linear recognition algorithms of weak-bisplit graphs and star1,2,3-free bipartite graphs. As a result of this algorithm, we present efficient solutions in this class of graphs for two optimization graph problems: the maximum balanced biclique problem and the maximum independent set problem.

Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.

An asymmetric container lemma and the structure of graphs with no induced $4$-cycle

The method of hypergraph containers, which was introduced several years ago by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph H , the threshold at which the distribution of edges in a typical H -free graph with a given number of edges undergoes a transition from ‘random-like’ to ‘structured’. On the other hand, for non-monotone hereditary graph properties, the standard version of this method does not allow one to establish even the existence of such a threshold. In this paper we introduce a refinement of the container method that takes into account the asymmetry between edges and non-edges in a sparse member of a hereditary graph property. As an application, we determine the approximate structure of a typical graph with n vertices, m edges, and no induced copy of the 4 -cycle, for each function m = m(n) satisfying n^{4/3}(\log n)^{4} \le m \ll n^{2} . We show that almost all such graphs G have the following property: the vertex set of G can be partitioned into an ‘almost independent’ set (a set with o(m) edges) and an ‘almost-clique’ (a set inducing a subgraph with density 1-o(1) ). The lower bound on m is optimal up to a polylogarithmic factor, as standard arguments show that if n \ll m \ll n^{4/3} , then almost all such graphs are ‘random-like’. As a further consequence, we deduce that the random graph G(n,p) conditioned to contain no induced 4 -cycles undergoes phase transitions at p = n^{-2/3 + o(1)} and p = n^{-1/3 + o(1)} .

A Spectral Extremal Problem on Non-Bipartite Triangle-Free Graphs

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda(G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $\lambda_1^2(G) + \lambda_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $\lambda (G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $\lambda (G) \le \beta (m)$, where $\beta(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

Some conditions for hamiltonian cycles in 1-tough (K2 ∪ kK1)-free graphs

Let k≥2 be an integer. We say that a graph G is (K2∪kK1)-free if it does not contain K2∪kK1 as an induced subgraph. Recently, Shi and Shan conjectured that every 1-tough and 2k-connected (K2∪kK1)-free graph is hamiltonian. In this paper, we solve this conjecture by proving that every 1-tough and k-connected (K2∪kK1)-free graph with minimum degree at least 3(k−1)2 is hamiltonian or the Petersen graph.

Extremal spectral results of planar graphs without vertex‐disjoint cycles

AbstractGiven a planar graph family , let and be the maximum size and maximum spectral radius over all ‐vertex ‐free planar graphs, respectively. Let be the disjoint union of copies of ‐cycles, and be the family of vertex‐disjoint cycles without length restriction. Tait and Tobin determined that is the extremal spectral graph among all planar graphs with sufficiently large order , which implies the extremal graphs of both and for are . In this paper, we first determine and and characterize the unique extremal graph for , and sufficiently large . Second, we obtain the exact values of and , which solve a conjecture of Li for .

Near optimal colourability on hereditary graph families

In this paper, we initiate a systematic study on a new notion called near optimal colourability which is closely related to perfect graphs and the Lovász theta function. A graph family G is near optimal colourable if there is a constant number c such that every graph G∈G satisfies χ(G)≤max⁡{c,ω(G)}, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. The near optimal colourable graph families together with the Lovász theta function are useful for the study of the colouring problems for hereditary graph families. We investigate the near optimal colourability for (H1,H2)-free graphs. Our main result is an almost complete characterisation for the near optimal colourability for (H1,H2)-free graphs with two exceptional cases, one of which is the celebrated Gyárfás conjecture. As an application of our results, we show that the colouring problem for (2K2,P4∨Kn)-free graphs is polynomial time solvable, which solves an open problem in Dabrowski and Paulusma (2018) [6].

Spanning trees of $$K_{1,4}$$-free graphs whose reducible stems have few leaves

Spanning trees of $$K_{1,4}$$-free graphs whose reducible stems have few leaves

Minimum degree stability of C2k+1 ${C}_{2k+1}$‐free graphs

AbstractWe consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a ‐free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist showed that for , if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist also pointed out that this result cannot be extended to . In this paper, we give a complete answer for any . We show that if and is an ‐vertex ‐free graph with , then is bipartite, and the bound is tight.

Linear χ-binding functions for some classes of (P 3∪P 2)-free graphs

The class of 2 K 2 -free graphs has been well studied in the past. In the paper “On the chromatic number of 2 K 2 -free graphs, Discrete Applied Mathematics, 253 (2019), 14–24”, it was shown that the class of { 2 K 2 , 2 K 1 + K p } -free graphs and { 2 K 2 , ( K 1 ∪ K 2 ) + K p } -free graphs admit a linear χ-binding function. In this paper, we study some subclasses of ( P 3 ∪ P 2 ) -free graphs which is a superclass of 2 K 2 -free graphs. We show that { P 3 ∪ P 2 , 2 K 1 + K p } -free graphs and { P 3 ∪ P 2 , ( K 1 ∪ K 2 ) + K p } -free graphs also admit linear χ-binding functions. In addition, we give a tight χ-binding function for { P 3 ∪ P 2 , HVN } -free graphs and improve the χ-bound for { P 3 ∪ P 2 , diamond } -free graphs with ω = 4.

Spectral Extremal Problem on Disjoint Color-Critical Graphs

For a given graph $F$, we say that a graph $G$ is $F$-free if it does not contain $F$ as a subgraph. A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. Let $K_r^{+}(a_1, a_2, \ldots, a_r)$ be the graph obtained from complete $r$-partite graph with parts of sizes $a_1\geqslant 2, a_2, \ldots, a_r$, by adding an edge to the first part. In this paper, we focus on the spectral extrema of disjoint color-critical graphs. For fixed $t, a_1,\ldots, a_r\ (r\geqslant 2)$ and large enough $n$, we characterize the unique $n$-vertex $tK_r^+(a_1,\ldots,a_r)$-free graph having the largest spectral radius. Moreover, let $F_1,\ldots,F_t$ be $t$ disjoint color-critical graphs with the same chromatic number. We identify the unique $n$-vertex $\bigcup_{i=1}^tF_i$-free graph having the largest spectral radius for sufficiently large $n$. Consequently, we generalize the main results obtained by Ni, Wang and Kang [Electron. J. Combin. 30 (1) (2023), #P1.20] and by Fang, Zhai and Lin [arXiv:2302.03229v2].