For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erd\H{o}s-S\'{o}s conjecture, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1$, $$\operatorname{ex}(n,K_r,T)=a\binom{k-1}{r}+\binom{b}{r},$$ where $n=a(k-1)+b$ with $0\le b<k-1$. In this paper, we confirm the conjecture for $r\ge \left\lceil (2k-1)/3\right\rceil$ and characterize all extremal graphs.
We show that there exists $C\in\mathbb{N}$ such that $K_t$-minor free graphs are $Ct$-colorable. The proof was found by GPT-6 Astra, following the directions by the authors.
In this paper, we examine the structure of strongly regular graphs with parameters $\lambda = 1$ and $\mu = 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of s...
A clique transversal of a graph is a set of vertices intersecting every maximal clique. We prove that deciding whether a graph has an inclusion-wise minimal clique transversal of size at least $k$ is W[1]-hard when parameterized by $k$.
Pascal Gollin, Tesshu Hanaka, Ekkehard Köhler et al.· 0 citations
For every integer $k\ge2$, Chen and Raspaud conjectured that each graph $G$ with odd girth $\og(G)\ge2k+1$ and maximum average degree $\mad(G)<2+1/k$ has a $(2k+1:k)$-coloring. In this paper, we prove the conjecture.
We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.
We show that if an n-vertex 5-connected graph has at least 4n-10 edges, then for any choice of five of its vertices, we can contract disjoint connected subgraphs containing these vertices to obtain $K_5$ as a minor. The bound on the number of edges is the best possible.
Z. Dvořák· 2 citations· ⚡1
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