Related papers: Tree independence number III. Thetas, prisms and s…

For two graphs $J$ and $H$, the generalized Tur\'{a}n number, denoted by $ex(n,J,H)$, is the maximum number of copies of $J$ in an $H$-free graph of order $n$. A linear forest $F$ is the disjoint union of paths. In this paper, we determine…

Let $G$ be a graph, $S \subseteq V(G)$ be a vertex set in $G$ and $r$ be a positive integer. The distance $r$-independence number of $S$ is the size of the largest subset $I \subseteq S$ such that no pair $u$, $v$ of vertices in $I$ have a…

This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

We introduce a ``Kirchhoff--Tur\'an'' variant of the extremal $C_4$ problem: among all simple connected $n$-vertex $C_4$-free graphs $G$, maximize the number of spanning trees $\tau(G)$. For the projective-plane orders $n=q^2+q+1$ we…

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of…

We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^\gamma$ where $\gamma\leq 1$ and $U$ is the uniform…

Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all $n \ge 6$, the maximum…

We study how the relationship between non-equivalent width parameters changes once we restrict to some special graph class. As width parameters, we consider treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence…

A $\Lambda$-tree is a $\Lambda$-metric space satisfying three axioms (1), (2) and (3). We give a characterization of those ordered abelian groups $\Lambda$ for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for…

For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most…

A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…

We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornu\'ejols, Kapoor and Vu\v skovi\'c, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]. In fact, the…

We introduce a lower bound for the independence number of an arbitrary $k$-uniform hypergraph that only depends on the number of vertices and number of edges of the hypergraph.

Every $n$-vertex planar triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$. This was first conjectured by Albertson, Bollob\'as and Tucker, and was later proved by Heckman and Thomas.…

Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $\delta(G) \alpha(G)\leq \Delta(G) \mu(G)$ for every graph $G$ with minimum degree $\delta(G)$, maximum degree $\Delta(G)$, independence number $\alpha(G)$, and matching number…

The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with…

We prove that every simplicial automorphism of the free splitting graph of a free group of at least rank 3 is induced by an outer automorphism of the free group.

We prove an upper bound for the independence number of a graph in terms of the largest Laplacian eigenvalue, and of a certain induced subgraph. Our bound is a refinement of a well-known Hoffman-type bound.

Given a graph $G$, the \textit{independence complex} $I(G)$ is the simplicial complex whose faces are the independent sets of $V(G)$. Let $\tilde{b}_i$ denote the $i$-th reduced Betti number of $I(G)$, and let $b(G)$ denote the sum of…