Free Graphs Research Articles

New results for MaxCut in H$H$‐free graphs

AbstractThe MaxCut problem asks for the size of a largest cut in a graph . It is well known that for any ‐edge graph , and the difference is called the surplus of . The study of the surplus of ‐free graphs was initiated by Erdős and Lovász in the 70s, who, in particular, asked what happens for triangle‐free graphs. This was famously resolved by Alon, who showed that in the triangle‐free case the surplus is , and found constructions matching this bound. We prove several new results in this area. Confirming a conjecture of Alon, Krivelevich and Sudakov, we show that for every fixed odd , any ‐free graph with edges has surplus . This is tight, as is shown by a construction of pseudo‐random ‐free graphs due to Alon and Kahale. It improves previous results of several researchers, and complements a result of Alon, Krivelevich and Sudakov which is the same bound when is even. Generalising the result of Alon, we allow the graph to have triangles, and show that if the number of triangles is a bit less than in a random graph with the same density, then the graph has large surplus. For regular graphs, our bounds on the surplus are sharp. We prove that an ‐vertex graph with few copies of and average degree has surplus , which is tight when is close to provided that a conjectured dense pseudo‐random ‐free graph exists. This result is used to improve the best known lower bound (as a function of ) on the surplus of ‐free graphs. Our proofs combine techniques from semi‐definite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

Bounds on Ramsey games via alterations

AbstractWe present a refinement of the classical alteration method for constructing ‐free graphs for suitable edge‐probabilities , we show that removing all edges in ‐copies of the binomial random graph does not significantly change the independence number. This differs from earlier alteration approaches of Erdős and Krivelevich, who obtained similar guarantees by removing one edge from each ‐copy (instead of all of them). We demonstrate the usefulness of our refined alternation method via two applications to online graph Ramsey games, where it enables easier analysis.

A refinement on the structure of vertex-critical (P5, gem)-free graphs

We give a new, stronger proof that there are only finitely many k-vertex-critical (P5, gem)-free graphs for all k. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all 6- and 7-vertex-critical (P5, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the k-colourability of (P5, gem)-free graphs for all k where the certificate is either a k-colouring or a (k+1)-vertex-critical induced subgraph. Our complete lists for k≤7 allow for the implementation of these algorithms for all k≤6.

Paths are Turán-good

We show that among K_{k+1}-free n-vertex graphs, the Turán graph contains the most copies of any path.

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

AbstractLet be a class of graphs with a membership test, , and let be the class of graphs in of path‐width at most . We present an interactive framework that finds an unavoidable set for , which is a set of graphs such that any graph in contains an isomorphic copy of a graph in . At the core of our framework is an algorithm that verifies whether a set of graphs is, indeed, unavoidable for . While obstruction sets are well‐studied, so far there is no general theory or algorithm for finding unavoidable sets. In general, it is undecidable whether a finite set of graphs is unavoidable for a given graph class. However, we give a criterion for termination: our algorithm terminates whenever is locally checkable of bounded maximum degree and is a finite set of connected graphs. For example, ‐regular graphs, ‐colourable graphs, and ‐free graphs are locally checkable classes. We put special emphasis on the case that is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high‐degree‐first path‐decompositions, which enables highly efficient pruning techniques. We exploit our framework to prove a new lower bound on the path‐width of cubic graphs. Moreover, we determine the extremal girth values of cubic graphs of path‐width for all and all smallest graphs which take on these extremal girth values. Further, we present a new constructive characterisation of the extremal cubic graphs of path‐width 3 and girth 4.

Extremal problems of double stars

In a generalized Tur\'an problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.

The maximum spectral radius of {C3,C5}-free graphs of given size

In this paper we consider the extremal problem on adjacency spectral radius of {C3,C5}-free graphs. Assume that G is a graph with m edges having no isolated vertices, and let λ be the spectral radius of its adjacency matrix. Firstly, by using the method of characterizing {C3,C5}-free non-bipartite graphs whose second largest eigenvalue is less than 54, we show that, if G is a {C3,C5}-free non-bipartite graph of size m, thenλ⩽∑u∈VGd(u)2−m+4f−54. Equality holds if and only if G≅C7, where d(u) is the degree of vertex u and f denotes the number of 4-cycles in G. Secondly, we show that, if G is a {C3,C5}-free non-bipartite graph of odd size m, then λ⩽θ(m) with equality if and only if G≅RK2,m−32, where θ(m) is the largest root ofx4−x3−(m−3)x2+(m−4)x+m−5=0 and RK2,m−32 is obtained by replacing an edge of the complete bipartite graph K2,m−32 with P5.

Turán number for odd‐ballooning of trees

AbstractThe Turán number is the maximum number of edges in an ‐free graph on vertices. Let be any tree. The odd‐ballooning of , denoted by , is a graph obtained by replacing each edge of with an odd cycle containing the edge, and all new vertices of the odd cycles are distinct. In this paper, we determine the exact value of for sufficiently large and being good, which generalizes all the known results on for being a star, due to Erdős, Füredi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018), and provides some counterexamples with chromatic number three to a conjecture of Keevash and Sudakov (2004), on the maximum number of edges not in any monochromatic copy of in a 2‐edge‐coloring of a complete graph of order .

Graph Colouring, Class Scheduling and Application in Modern Computer Science

Abstract: Course timetabling and exam timetabling are two most common academic scheduling problems in any of the educational institution. A desirable schedule combines resources in a way which satisfies essential and preferential constrains. In this paper, we focus on college course timetabling including hard and soft constraints. Our Objective is to inculcate graph colouring approach to obtain a conflict free course graph.

Some bounds on the size of maximum G-free sets in graphs

For a given graph [Formula: see text], the independence number [Formula: see text] of [Formula: see text] is the size of the maximum independent set of [Formula: see text]. Finding the maximum independent set in a graph is NP-hard. Another version of the independence number is defined as the size of the maximum-induced forest of [Formula: see text], and called the forest number of [Formula: see text], and denoted by [Formula: see text]. Finding [Formula: see text] is also an NP-hard problem. Suppose that [Formula: see text] is a graph, and [Formula: see text] is a family of graphs, a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring if there exists a decomposition of [Formula: see text] into sets [Formula: see text], [Formula: see text], so that [Formula: see text] for each [Formula: see text], and each [Formula: see text]. [Formula: see text] is [Formula: see text]-free, if the subgraph of [Formula: see text] induced by [Formula: see text] is [Formula: see text]-free, i.e., it contains no copy of [Formula: see text]. Finding a maximum subset [Formula: see text] of [Formula: see text], such that [Formula: see text] is a [Formula: see text]-free graph is a very hard problem as well. In this paper, we study the generalized version of the independence number of a graph. Also, we give some bounds about the size of the maximum [Formula: see text]-free subset of graphs as another purpose of this paper.

Subgraph Densities in $K_r$-Free Graphs

In this paper we disprove a conjecture of Lidický and Murphy about the number of copies of a given graph in a $K_r$-free graph and give an alternative general conjecture. We also prove an asymptotically tight bound on the number of copies of any bipartite graph of radius at most~$2$ in a triangle-free graph.

On Aα spectral extrema of graphs forbidding even cycles

Given a graph G, for a real number α∈[0,1], Nikiforov (2017) proposed the Aα-matrix of G as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of Aα(G) is called the Aα-index of G. For n>k, let Sn,k be the join of a clique on k vertices with an independent set of n−k vertices. Then Sn,k+ denotes the graph obtained from Sn,k by adding an edge to connect two vertices of degree k. Very recently, Cioabǎ, Desai and Tait resolved the Nikiforov's conjecture: For fixed k≥2, and sufficiently large n, the {C2k+1,C2k+2}-free graph of order n with maximum adjacency spectral radius is Sn,k and the C2k+2-free graph of order n with maximum adjacency spectral radius is Sn,k+. In this paper, we confirm the Aα-spectral version of this conjecture: For fixed k≥2,0<α<1 and n≥324k6(k+1)2α6(1−α)2, the {C2k+1,C2k+2}-free graph of order n with maximum Aα-index is Sn,k and the C2k+2-free graph of order n with maximum Aα-index is Sn,k+.

On algorithmic applications of sim-width and mim-width of (H1,H2)-free graphs

Mim-width and sim-width are among the most powerful graph width parameters, with sim-width more powerful than mim-width, which is in turn more powerful than clique-width. While several NP-hard graph problems become tractable for graph classes whose mim-width is bounded and quickly computable, no algorithmic applications of boundedness of sim-width are known. In Kang et al. (2017) [32], it is asked whether Independent Set and 3-Colouring are NP-complete on graphs of sim-width at most 1. We observe that, for each k∈N, Listk-Colouring is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. Moreover, we show that if the same holds for Independent Set, then IndependentH-Packing is polynomial-time solvable for graph classes whose sim-width is bounded and quickly computable. This problem is a common generalisation of Independent Set, Induced Matching, Dissociation Set and k-Separator.We also make progress toward classifying the mim-width of (H1,H2)-free graphs in the case H1 is complete or edgeless. Our results solve some open problems in Brettell et al. (2022) [6].

Two Completely Independent Spanning Trees of $$P_4$$-Free Graphs

Two Completely Independent Spanning Trees of $$P_4$$-Free Graphs

A note on the bounds for the spectral radius of graphs

Let G=(V,E) be a finite undirected graph of order n and of size m. Let Δ and δ be the largest and the smallest degree of G, respectively. The spectral radius of G is the largest eigenvalue of the adjacency matrix of the graph G. In this note we give new bounds on the spectral radius of {C3,C4}-free graphs in terms of m,n,Δ and δ. Computer search shows that in most of the cases the bounds derived in this note are better than the existing bounds.

Remarks on component factors in K1,r-free graphs

An ℱ-factor is a spanning subgraph H such that each connected component of H is isomorphic to some graph in ℱ. We use Pk and K1,r to denote the path of order k and the star of order r + 1, respectively. In particular, H is called a {P2, P3}-factor of G if ℱ = {P2, P3}; H is called a P≥k-factor of G if ℱ = {Pk, Pk+1,…}, where k ≥ 2; H is called an Sn-factor of G if ℱ = {P2, P3, K1,3,…, K1,n}, where n ≥ 2. A graph G is called a ℱ≥k-factor covered graph if there is a ℱ≥k-factor of G including e for any e ∈ E(G). We call a graph G is K1,r-free if G does not contain an induced subgraph isomorphic to K1,r. In this paper, we give a minimum degree condition for the K1,r-free graph with an Sn-factor and the K1,r-free graph with a ℱ≥3-factor, respectively. Further, we obtain sufficient conditions for K1,r-free graphs to be ℱ≥2-factor, ℱ≥3-factor or {P2, P3}-factor covered graphs. In addition, examples show that our results are sharp.

All longest cycles in a 2‐connected partial 3‐tree share a common vertex

AbstractIn 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. In this paper, we study the related question on longest cycles instead of longest paths (to make the question valid, we require 2‐connectivity). The answer for cycles is also negative and not so much attention has been given to it. Classes for which the answer is known to be positive are dually chordal graphs and 3‐trees. Perhaps the main open class for the question of both paths and cycles is the class of chordal graphs. In this paper, we show that all longest cycles intersect in graphs with treewidth at most three. This class of graphs includes several important subclasses of graphs, such as series‐parallel graphs and ‐free chordal graphs.

Non-Empty Intersection of Longest Paths in $H$-Free Graphs

We make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. We also show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$, and its independence number $\alpha(G)$ satisfies $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.

Cops and robber on subclasses of P5-free graphs

The game of cops and robber is a turn-based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say that the team of cops wins the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in a connected graph is called the cop number of the graph. Sivaraman (2019) [16] conjectured that for every t≥5, the cop number of a connected Pt-free graph is at most t−3, where Pt denotes a path on t vertices. Turcotte (2022) [18] showed that the cop number of any 2K2-free graph is at most 2, which was earlier conjectured by Sivaraman and Testa. Note that if a connected graph is 2K2-free, then it is also P5-free. Liu showed that the cop number of a connected (Pt, H)-free graph is at most t−3, where H is a cycle of length at most t or a claw. So the conjecture of Sivaraman is true for (P5, H)-free graphs, where H is a cycle of length at most 5 or a claw. In this paper, we show that the cop number of a connected (P5,H)-free graph is at most 2, where H∈{diamond, paw, K4, 2K1∪K2, K3∪K1}.

Bipartite-ness under smooth conditions

Abstract Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such $G$ is called $\mathcal{F}$ -free). For $1\leq \beta \lt \alpha \lt 2$ , a family $\mathcal{F}$ of bipartite graphs is $(\alpha,\beta )$ -smooth if for some $\rho \gt 0$ and every $m\leq n$ , $z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$ . Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any $(\alpha,\beta )$ -smooth family $\mathcal{F}$ , there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$ , any $n$ -vertex $\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least $\rho (\frac{2n}{5}+o(n))^{\alpha -1}$ is bipartite. In this paper, we strengthen their result by showing that for every real $\delta \gt 0$ , there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$ , any $n$ -vertex $\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least $\delta n^{\alpha -1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $\mathcal{F}$ consisting of the single graph $K_{s,t}$ when $t\gg s$ . We also prove an analogous result for $C_{2\ell }$ -free graphs for every $\ell \geq 2$ , which complements a result of Keevash, Sudakov and Verstraëte.