Free Graphs Research Articles

The spectral Turán problem about graphs of given size with forbidden subgraphs

Let G be a graph of size m and ρ ( G ) be the spectral radius of its adjacency matrix. A graph is said to be H-free if it does not contain a subgraph isomorphic to H. Let C l 1 be the graph obtained by adding a pendent vertex to a cycle of length l. In this paper, we determine the graphs with the maximum spectral radius over C 3 1 -free graphs, and C 4 1 -free graphs with m ≥ 9 .

Spectral Extremal Problem on $t$ Copies of $\ell$-Cycles

Extremal problem on cycles plays an important role in extremal graph theory. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\geq f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erdős [Arch. Math. 13 (1962) 222–227] as well as the research on $ex(n,C_{2\ell+1})$ of Füredi and Gunderson [Combin. Probab. Comput. 24 (2015) 641–645]. On the other hand, motivated by the spectral Turán-type problem proposed by Nikiforov, we obtain the extremal spectral radius $spex(n,tC_{\ell})$ for any fixed $t,\ell$ and large enough $n$. Our results extend some classic spectral extremal results or conjectures on odd cycles and even cycles. Our results also give some inspirations for general spectral Turán-type problem $spex(n,F)$ on bipartite or non-partite $F$.

Thermodynamics-consistent graph neural networks.

We propose excess Gibbs free energy graph neural networks (GE-GNNs) for predicting composition-dependent activity coefficients of binary mixtures. The GE-GNN architecture ensures thermodynamic consistency by predicting the molar excess Gibbs free energy and using thermodynamic relations to obtain activity coefficients. As these are differential, automatic differentiation is applied to learn the activity coefficients in an end-to-end manner. Since the architecture is based on fundamental thermodynamics, we do not require additional loss terms to learn thermodynamic consistency. As the output is a fundamental property, we neither impose thermodynamic modeling limitations and assumptions. We demonstrate high accuracy and thermodynamic consistency of the activity coefficient predictions.

The Chromatic Number of (P5, HVN)-free Graphs

The Chromatic Number of (P5, HVN)-free Graphs

Spanning Trees with at most $5$ Leaves and Branch Vertices in Total of $K_{1,5}$-free Graphs

Spanning Trees with at most $5$ Leaves and Branch Vertices in Total of $K_{1,5}$-free Graphs

Variants of the Gyárfás–Sumner conjecture: Oriented trees and rainbow paths

AbstractGiven a finite family of graphs, we say that a graph is “‐free” if does not contain any graph in as a subgraph. We abbreviate ‐free to just “‐free” when . A vertex‐colored graph is called “rainbow” if no two vertices of have the same color. Given an integer and a finite family of graphs , let denote the smallest integer such that any properly vertex‐colored ‐free graph having contains an induced rainbow path on vertices. Scott and Seymour showed that exists for every complete graph . A conjecture of N. R. Aravind states that . The upper bound on that can be obtained using the methods of Scott and Seymour setting are, however, super‐exponential. Gyárfás and Sárközy showed that . For , we show that and therefore, . This significantly improves Gyárfás and Sárközy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that , where . Moreover, in each case, our results imply the existence of at least distinct induced rainbow paths on vertices. Along the way, we obtain some new results on an oriented variant of the Gyárfás–Sumner conjecture. For , let denote the orientations of in which one vertex has out‐degree or in‐degree . We show that every ‐free oriented graph having a chromatic number at least and every bikernel‐perfect oriented graph with girth having a chromatic number at least contains every oriented tree on at most vertices as an induced subgraph.

Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (P6+sP2)-free graphs for every s≥0, extending a known result for P5-free graphs (Bouquet and Picouleau, 2020).

Reducing linear Hadwiger’s conjecture to coloring small graphs

In 1943, Hadwiger conjectured that every graph with no K t K_t minor is ( t − 1 ) (t-1) -colorable for every t ≥ 1 t\ge 1 . In the 1980s, Kostochka and Thomason independently proved that every graph with no K t K_t minor has average degree O ( t log ⁡ t ) O(t\sqrt {\log t}) and hence is O ( t log ⁡ t ) O(t\sqrt {\log t}) -colorable. Recently, Norin, Song and the second author showed that every graph with no K t K_t minor is O ( t ( log ⁡ t ) β ) O(t(\log t)^{\beta }) -colorable for every β > 1 / 4 \beta > 1/4 , making the first improvement on the order of magnitude of the O ( t log ⁡ t ) O(t\sqrt {\log t}) bound. The first main result of this paper is that every graph with no K t K_t minor is O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. This is actually a corollary of our main technical result that the chromatic number of a K t K_t -minor-free graph is bounded by O ( t ( 1 + f ( G , t ) ) ) O(t(1+f(G,t))) where f ( G , t ) f(G,t) is the maximum of χ ( H ) a \frac {\chi (H)}{a} over all a ≥ t log ⁡ t a\ge \frac {t}{\sqrt {\log t}} and K a K_a -minor-free subgraphs H H of G G that are small (i.e. O ( a log 4 ⁡ a ) O(a\log ^4 a) vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small K t K_t -minor-free graphs, we show that K t K_t -minor-free graphs are O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. Second, it shows that proving Linear Hadwiger’s Conjecture (that K t K_t -minor-free graphs are O ( t ) O(t) -colorable) reduces to proving it for small graphs. Third, we prove that K t K_t -minor-free graphs with clique number at most log ⁡ t / ( log ⁡ log ⁡ t ) 2 \sqrt {\log t}/ (\log \log t)^2 are O ( t ) O(t) -colorable. This implies our final corollary that Linear Hadwiger’s Conjecture holds for K r K_r -free graphs for every fixed r r ; more generally, we show there exists C ≥ 1 C\ge 1 such that for every r ≥ 1 r\ge 1 , there exists t r t_r such that for all t ≥ t r t\ge t_r , every K r K_r -free K t K_t -minor-free graph is C t Ct -colorable. One key to proving the main theorem is building the minor in two new ways according to whether the chromatic number ‘separates’ in the graph: sequentially if the graph is the chromatic-inseparable and recursively if the graph is chromatic-separable. The other key is a new standalone result that every K t K_t -minor-free graph of average degree d = Ω ( t ) d=\Omega (t) has a subgraph on O ( t log 3 ⁡ t ) O(t \log ^3 t) vertices with average degree Ω ( d ) \Omega (d) .

Stability from graph symmetrization arguments in generalized Turán problems

AbstractGiven graphs and , denotes the largest number of copies of in ‐free ‐vertex graphs. Let . We say that is F‐Turán‐stable if the following holds. For any there exists such that if an ‐vertex ‐free graph contains at least copies of , then the edit distance of and the ‐partite Turán graph is at most . We say that is weakly F‐Turán‐stable if the same holds with the Turán graph replaced by any complete ‐partite graph . It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most are weakly ‐Turán‐stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph , if is large enough, then is ‐Turán‐stable. Finally, we prove that if is bipartite, then it is weakly ‐Turán‐stable for large enough.

Directed graphs with lower orientation Ramsey thresholds

We investigate the threshold p⃗H = p⃗H (n) for the Ramsey-type property G(n, p) → ⃗H, where G(n, p) is the binomial random graph and G → ⃗H indicates that every orientation of the graph G contains the oriented graph ⃗ H as a subdigraph. Similarly to the classical Ramsey setting, the upper bound p⃗H ⩽ Cn−1/m2(⃗ H) is known to hold for some constant C = C( ⃗ H), where m2(⃗H) denotes the maximum 2-density of the underlying graph H of ⃗ H. While this upper bound is indeed the threshold for some ⃗H, this is not always the case. We obtain examples arising from rooted products of orientations of sparse graphs (such as forests, cycles and, more generally, subcubic {K3, K3,3}-free graphs) and arbitrarily rooted transitive triangles.

Good r-divisions Imply Optimal Amortized Decremental Biconnectivity

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(m+n)$$\\end{document} time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

Borodin–Kostochka conjecture holds for [formula omitted]-free graphs

The Borodin–Kostochka conjecture says that for a graph G, if Δ(G)≥9, then χ(G)≤max{Δ(G)−1,ω(G)}. Cranston and Rabern in [SIAM J. Discrete. Math. 27 (2013) 534–549] proved the conjecture holding for K1,3-free graphs. In this paper, we prove that the conjecture holds for K1,3¯-free graphs, where K1,3¯ denotes the complement of K1,3.

χ-binding function for (C4, t-broom+)-free graphs

For a positive integer t, a t-broom+ is a graph obtained from K1,t+1 by subdividing an edge twice. In this paper, we prove that the Ramsey number φ(n,ω) in the family of C4-free graphs is linear in ω and obtain an O(ω2)χ-binding function for the family of (C4,t-broom+)-free graphs. This generalizes some results of Liu et al. [13].

Structure of some ( [formula omitted], [formula omitted])-free graphs with application to colorings

We use Pt and Ct to denote a path and a cycle on t vertices, respectively. A diamond is a graph obtained from two triangles that share exactly one edge. A kite is a graph that consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A gem is a graph that consists of a P4 plus a vertex adjacent to all vertices of the P4. Choudum et al. (2021) proved that every (P7,C7,C4, gem)-free graph has either a clique cutset or a vertex whose neighbors is the union of two cliques unless it is a clique blowup of the Petersen graph, and every non-Petersen (P7,C7,C4, diamond)-free graph has either a clique cutset or has a vertex of degree at most max{2,ω(G)−1}. In this paper, we extend these two results respectively to (P7,C4, gem)-free graphs and (P7,C4, diamond)-free graphs. We also prove that if G is a (P7,C4, kite)-free graph with δ(G)≤ω(G) and without clique cutsets, then there is an integer ℓ≥0 such that G=Kℓ+H, where H is either the Petersen graph or a well-defined graph F. This implies a polynomial algorithm which can determine the chromatic number of (P7,C4, diamond)-free graphs, color (P7,C4, kite)-free graphs G with (ω(G)+1) colors, and color (P7,C4, gem)-free graphs G with (2ω(G)−1) colors.

Ramsey‐type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture

AbstractGyárfás and Sumner independently conjectured that for every tree , there exists a function such that every ‐free graph satisfies , where and are the chromatic number and the clique number of , respectively. This conjecture gives a solution of a Ramsey‐type problem on the chromatic number. For a graph , the induced SP‐cover number (resp. the induced SP‐partition number ) of is the minimum cardinality of a family of induced subgraphs of such that each element of is a star or a path and (resp. ). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey‐type problems for two invariants and , which are analogies of the Gyárfás‐Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey‐type problems for widely studied invariants.

Edge DP-Coloring in K4-Minor Free Graphs and Planar Graphs

The edge DP-chromatic number of G, denoted by χDP′(G), is the minimum k such that G is edge DP-k-colorable. In 1999, Juvan, Mohar, and Thomas proved that the edge list chromatic number of K4-minor free graph G with Δ≥3 is Δ. In this paper, we prove that if G is a K4-minor free graph, then χDP′(G)∈{Δ,Δ+1}, and equality χDP′(G)=Δ+1 holds for some K4-minor free graph G with Δ=3. Moreover, if G is a planar graph with Δ≥9 and with no intersecting triangles, then χDP′(G)=Δ.

Multiple Views to Free Graph Augmentations

Multiple Views to Free Graph Augmentations

On the typical structure of graphs not containing a fixed vertex‐critical subgraph

AbstractThis work studies the typical structure of sparse ‐free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph . Extending the seminal result of Osthus, Prömel, and Taraz that addressed the case where is an odd cycle, Balogh, Morris, Samotij, and Warnke proved that, for every , the structure of a random ‐free graph with vertices and edges undergoes a phase transition when crosses an explicit (sharp) threshold function . They conjectured that a similar threshold phenomenon occurs when is replaced by any strictly 2‐balanced, edge‐critical graph . In this paper, we resolve this conjecture. In fact, we prove that the structure of a typical ‐free graph undergoes an analogous phase transition for every in a family of vertex‐critical graphs that includes all edge‐critical graphs.

Spectral Extremal Results on Trees

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where '$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228-2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq \rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

Turán number of the odd‐ballooning of complete bipartite graphs

AbstractGiven a graph , the Turán number is the maximum possible number of edges in an ‐vertex ‐free graph. The study of Turán number of graphs is a central topic in extremal graph theory. Although the celebrated Erdős‐Stone‐Simonovits theorem gives the asymptotic value of for nonbipartite , it is challenging in general to determine the exact value of for . The odd‐ballooning of is a graph such that each edge of is replaced by an odd cycle and all new vertices of odd cycles are distinct. Here the length of odd cycles is not necessarily equal. The exact value of Turán number of the odd‐ballooning of is previously known for being a cycle, a path, a tree with assumptions, and . In this paper, we manage to obtain the exact value of Turán number of the odd‐ballooning of with , where and each odd cycle has length at least five.