Free Graphs Research Articles

Depth-defying OOF-GNN: Sailing smoothly amidst GNN waves

When it comes to machine learning on graphs, Graph Neural Networks(GNNs) is a potent tool. By iteratively propagating neural messages along the edges of the input graph, GNNs integrate node feature information with graph structure. Nonetheless, not all neighborhood data is valuable. Some of the neighbor data may not be necessary for the downstream task. For a controlled and relevant information aggregation, we propose a flexible method OOF-GNN (Over-smoothing and Over-squashing Free Graph Neural Network). The main notion of this technique is to limit the impact of irrelevant and excessive information propagation during message passing. An extensive discussion of the theoretical and experimental justifications, associated with this method is presented. This technique works on all baseline models (e.g. GCN, GraphSAGE, and GAT). It allows them to gather information from the K-hop neighborhood without node information being over-smoothed and over-squashed. The efficacy of OFF-GNN in addressing the challenges of under-reaching, over-smoothing, and over-squashing, as well as improving the performance of GNN models in node classification tasks, has been demonstrated through comprehensive experimentation on both academic benchmarks and real-world datasets.

Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a P_5-free graph with clique number omega ge 3 has chromatic number at most omega ^{log _2(omega )}. The best previous result was an exponential upper bound (5/27)3^{omega }, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for P_5, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P_5-free graphs, and our result is an attempt to approach that.

The one-visibility localization game

We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number ζ1. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for ζ1 on k-ary trees with k≥2 that differ by a multiplicative constant, showing that the parameter is unbounded on k-ary trees. We provide a O(n) bound for Kh-minor free graphs of order n, and we show Cartesian grids meet this bound by determining their one-visibility localization number up to four values. We present upper bounds on ζ1 using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.

Coloring [formula omitted]-free graphs with [formula omitted] colors

A house is the graph that consists of an induced 4-vertex cycle and a single vertex with precisely two adjacent neighbors on the cycle. The Borodin–Kostochka Conjecture states that for each graph G with Δ(G)≥9, we have χ(G)≤max{Δ(G)−1,ω(G)}. We show that this conjecture holds for {P2∪P3,house}-free graphs.

Forbidden pairs of disconnected graphs for supereulerianity of connected graphs

Let H be a set of graphs. A graph G is said to be H-free if G does not contain H as an induced subgraph for all H∈H, and we call H a forbidden pair if |H|=2. A graph is called supereulerian if it contains a spanning connected even subgraph. In 1979, Pulleyblank showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. In this paper, we characterize all pairs of graphs R,S (not necessary connected) such that every 2-connected or 2-edge-connected {R,S}-free graph (of sufficiently large order) is supereulerian. We also characterize all minimal 2-connected non-supereulerian graphs, which extends a result by the third author and Xiong (2017) [29].

Existence of 4-factors in star-free graphs with edge-connectivity conditions

Let t ≥ 3 be an integer. We show that every 2-edge-connected K 1 , t -free graph with minimum degree at least max { 2 t − 3 , ⌈ 3 t + 1 2 ⌉ } , as well as every 3-edge-connected K 1 , t -free graph with minimum degree at least ⌈ 3 t + 1 2 ⌉ , has a 4-factor.

Connected [formula omitted]-free graphs whose second largest eigenvalue does not exceed 1

Connected graphs whose second largest eigenvalue λ2 does not exceed 1 have been investigated in the last four decades. Over the years only few particular classes with this spectral property are completely determined. For example, they include bipartite graphs, line graphs and threshold graphs. In this paper we determine all connected (K4−e)-free graphs, i.e., the graphs that do not contain an induced subgraph isomorphic to the graph obtained by removing an edge from the complete graph of order 4; such graphs are also called diamond-free graphs. Since every triangle-free graph is automatically diamond-free, our results includes all triangle-free graphs with λ2≤1. Moreover, it includes all connected bipartite graphs with the same spectral property, and therefore strengthens the result of Petrović (J. Combin. Theory B, 1991).

Vertex-critical [formula omitted]-free graphs

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. A Pt is the path on t vertices. A chair is a P4 with an additional vertex adjacent to one of the middle vertices of the P4. A graph G is k-vertex-critical if G has chromatic number k but every proper induced subgraph of G has chromatic number less than k. In this paper, we prove that there are only finitely many 5-vertex-critical (P5,chair)-free graphs.

Optimal chromatic bound for (P2+P3,P2+P3¯ ${P}_{2}+{P}_{3},\bar{{P}_{2}+{P}_{3}}$)‐free graphs

AbstractFor a graph , let () denote its chromatic (clique) number. A is the graph obtained by taking the disjoint union of a two‐vertex path and a three‐vertex path . A is the complement graph of a . In this paper, we study the class of ()‐free graphs and show that every such graph with satisfies . Moreover, the bound is tight. Indeed, for any and , there is a ()‐free graph with and .

Clique‐width: Harnessing the power of atoms

AbstractMany NP‐complete graph problems are polynomial‐time solvable on graph classes of bounded clique‐width. Several of these problems are polynomial‐time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut‐set) of . Hence, we initiate a systematic study into boundedness of clique‐width of atoms of hereditary graph classes. A graph is ‐free if is not an induced subgraph of , and it is ‐free if it is both ‐free and ‐free. A class of ‐free graphs has bounded clique‐width if and only if its atoms have this property. This is no longer true for ‐free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique‐width on ‐free atoms for all but 18 cases.

Hat Guessing Numbers of Strongly Degenerate Graphs

Assume players are placed on vertices of a graph . The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a color out of available colors. The players can see the hat of each of their neighbors in but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer such that there is a winning strategy for the players is denoted by , and this is called the hat guessing number of . Although this game has received much attention in recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from to 40 and further derive upper bounds on the hat guessing number for any class of -free graphs with bounded expansion, such as the class of -free planar graphs; more generally, for -free graphs with bounded Hadwiger number or without a -subdivision; and for Erdős–Rényi random graphs with constant average degree.

Induced Subgraphs and Tree Decompositions IV. (Even Hole, Diamond, Pyramid)-Free Graphs

A hole in a graph $G$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $K_4$ by removing an edge. A pyramid is a graph consisting of a vertex $a$ called the apex and a triangle $\{b_1, b_2, b_3\}$ called the base, and three paths $P_i$ from $a$ to $b_i$ for $1 \leq i \leq 3$, all of length at least one, such that for $i \neq j$, the only edge between $P_i \setminus \{a\}$ and $P_j \setminus \{a\}$ is $b_ib_j$, and at most one of $P_1$, $P_2$, and $P_3$ has length exactly one. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Cameron, da Silva, Huang, and Vušković proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, pyramid, $K_4$)-free graphs of arbitrarily large treewidth. Here, we show that for every $t$, (even hole, pyramid, diamond, $K_t$)-free graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon contain diamonds, so our result is sharp in the sense that it is false if we do not exclude diamonds. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses “non-crossing decompositions” methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of “non-crossing decompositions” to prove bounded treewidth in a graph class of unbounded maximum degree.

A note on pseudorandom Ramsey graphs

For fixed s \ge 3 , we prove that if optimal K_s -free pseudorandom graphs exist, then the Ramsey number r(s,t) is t^{s-1+o(1)} as t \rightarrow \infty . Our method also improves the best lower bounds for r(C_{\ell},t) obtained by Bohman and Keevash from the random C_{\ell} -free process by polylogarithmic factors for all odd \ell \geq 5 and \ell \in \{6,10\} . For \ell = 4 it matches their lower bound from the C_4 -free process. We also prove, via a different approach, that r(C_5, t)> (1+\nobreak o(1))t^{11/8} and r(C_7, t)> (1+\nobreak o(1))t^{11/9} . These improve the exponent of t in the previous best results and appear to be the first examples of graphs F with cycles for which such an improvement of the exponent for r(F, t) is shown over the bounds given by the random F -free process and random graphs.

Universal graphs for the topological minor relation

AbstractA subgraph‐universal graph/a topological minor‐universal graph in a class of graphs is a graph in which contains every graph in as a subgraph/topological minor. We prove that the class of all countable planar graphs does not contain a topological minor‐universal graph. This answers a question of Diestel and Kühn and strengthens a result of Pach stating that there is no subgraph‐universal graph in . Furthermore, we characterise for which subdivided stars there is a topological minor‐universal graph in the class of all countable ‐free graphs.

Strong Edge Coloring of K4(t)-Minor Free Graphs

A strong edge coloring of a graph G is a proper coloring of edges in G such that any two edges of distance at most 2 are colored with distinct colors. The strong chromatic index χs′(G) is the smallest integer l such that G admits a strong edge coloring using l colors. A K4(t)-minor free graph is a graph that does not contain K4(t) as a contraction subgraph, where K4(t) is obtained from a K4 by subdividing edges exactly t−4 times. The paper shows that every K4(t)-minor free graph with maximum degree Δ(G) has χs′(G)≤(t−1)Δ(G) for t∈{5,6,7} which generalizes some known results on K4-minor free graphs by Batenburg, Joannis de Verclos, Kang, Pirot in 2022 and Wang, Wang, and Wang in 2018. These upper bounds are sharp.

On Some Generalized Vertex Folkman Numbers

For a graph G and integers $$a_i\ge 1$$ , the expression $$G \rightarrow (a_1,\ldots ,a_r)^v$$ means that for any r-coloring of the vertices of G there exists a monochromatic $$a_i$$ -clique in G for some color $$i \in \{1,\ldots ,r\}$$ . The vertex Folkman numbers are defined as $$F_v(a_1,\ldots ,a_r;H) = \min \{|V(G)| : G$$ is H-free and $$G \rightarrow (a_1,\ldots ,a_r)^v\}$$ , where H is a graph. Such vertex Folkman numbers have been extensively studied for $$H=K_s$$ with $$s>\max \{a_i\}_{1\le i \le r}$$ . If $$a_i=a$$ for all i, then we use notation $$F_v(a^r;H)=F_v(a_1,\ldots ,a_r;H)$$ . Let $$J_k$$ be the complete graph $$K_k$$ missing one edge, i.e. $$J_k=K_k-e$$ . In this work we focus on vertex Folkman numbers with $$H=J_k$$ , in particular for $$k=4$$ and $$a_i\le 3$$ . A result by Nešetřil and Rödl from 1976 implies that $$F_v(3^r;J_4)$$ is well defined for any $$r\ge 2$$ . We present a new and more direct proof of this fact. The simplest but already intriguing case is that of $$F_v(3,3;J_4)$$ , for which we establish the upper bound of 135 by using the $$J_4$$ -free process. We obtain the exact values and bounds for a few other small cases of $$F_v(a_1,\ldots ,a_r;J_4)$$ when $$a_i \le 3$$ for all $$1 \le i \le r$$ , including $$F_v(2,3;J_4)=14$$ , $$F_v(2^4;J_4)=15$$ , and $$22 \le F_v(2^5;J_4) \le 25$$ . Note that $$F_v(2^r;J_4)$$ is the smallest number of vertices in any $$J_4$$ -free graph with chromatic number $$r+1$$ . Most of the results were obtained with the help of computations, but some of the upper bound graphs we found are interesting by themselves.

Balanced Subdivisions of Cliques in Graphs

Given a graph H, a balanced subdivision of H is a graph obtained from H by subdividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer $$k\ge 1$$ , high average degree is sufficient to guarantee a balanced subdivision of $$K_k$$ . Recently, Liu and Montgomery resolved this conjecture.We give an optimal estimate up to an absolute constant factor by showing that there exists $$c>0$$ such that for sufficiently large d, every graph with average degree at least d contains a balanced subdivision of a clique with at least $$cd^{1/2}$$ vertices. It also confirms a conjecture from Verstraëte: every graph of average degree $$cd^2$$ , for some absolute constant $$c>0$$ , contains a pair of disjoint isomorphic subdivisions of the complete graph $$K_d$$ . We also prove that there exists some absolute $$c>0$$ such that for sufficiently large d, every $$C_4$$ -free graph with average degree at least d contains a balanced subdivision of the complete graph $$K_{cd}$$ , which extends a result of Balogh, Liu and Sharifzadeh.

Generating $I$-Eigenvalue Free Threshold Graphs

A graph is said to be $I$-eigenvalue free if it has no eigenvalues in the interval $I$ with respect to the adjacency matrix $A$. In this paper we present twoalgorithms for generating $I$-eigenvalue free threshold graphs.

Improved bounds on the chromatic number of (P5, flag)-free graphs

Given a positive integer t, let Pt and Kt respectively denote the chordless path and the complete graph on t vertices. For a graph G, let χ(G) and ω(G) respectively denote the chromatic number and clique number of G. It is known that every (P5,K4)-free graph G satisfies χ(G)≤5, and the bound is tight. A flag is the graph obtained from a K4 by attaching a pendent vertex. Clearly, the class of flag-free graphs generalizes the class of K4-free graphs. In this paper, we show the following:•Every (P5,flag,K5)-free graph G that contains a K4 satisfies χ(G)≤8.•Every (P5,flag,K6)-free graph G satisfies χ(G)≤8.•Every (P5,flag,K7)-free graph G satisfies χ(G)≤9. We also give examples to show that the given bounds are tight. Further, we show that every (P5, flag)-free graph G with ω(G)≥4 satisfies χ(G)≤max⁡{8,2ω(G)−3}, and the bound is tight for ω(G)∈{4,5,6}. We note that our bound is an improvement over that given in Dong et al. [3,4].

Polynomial χ-binding functions for t-broom-free graphs

For any positive integer t, a t-broom is a graph obtained from K1,t+1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t-brooms, we have χ(G)=o(ω(G)t+1), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. When t=2, this answers a question of Schiermeyer and Randerath. Moreover, for t=2, we strengthen the bound on χ(G) to 7ω(G)2, confirming a conjecture of Sivaraman. For t≥3 and {t-broom, Kt,t}-free graphs, we improve the bound to o(ωt).