11 problems

Let $F=kP_{\ell}$ be a linear forest consisting of $k\geq 2$ vertex-disjoint paths of length $\ell$, with $\ell\neq 3$. Let $G_F(n)$, $\delta_F$, and $\mu$ be as in Theorem BK, whe…

Let $G$ be a $k$-connected graph with $k\geq 2$, and let $F$ be a linear forest, meaning a graph whose components are paths, that is a subgraph of $G$ with $l$ edges and $t$ isolat…

Let $G$ be a finite planar graph of order $n$. An induced linear forest is an induced subgraph that is a forest consisting only of paths. Chappell's conjecture. The graph $G$ has a…

Let $n$, $k$, and $s$ be integers, and let $P_3$ denote the path on three vertices. Write $kP_3$ for the disjoint union of $k$ copies of $P_3$, and let $M_n$ be the graph consistin…

All graphs are finite, simple, and undirected; for a graph $G$, let $V(G)$ be its vertex set and let $d(v)$ denote the degree of $v$. A linear forest is a forest in which every com…

Let $r\geq 2$, and let $\mathcal{L}_{n,k}^{(r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. Here a tight linear forest is an $r$-graph who…

A linear forest is a disjoint union of paths, and a linear forest is $\\{P_2,P_3\\}$-free when none of its components is isomorphic to $P_2$ or $P_3$. An antimagic labeling assigns…

Let $P_3$ be the path on three vertices, and let $gr_k(K_3:n_1P_3,\ldots,n_kP_3)$ denote the corresponding Gallai-Ramsey number. For positive integers $k,n_1,n_2,\ldots,n_k$ with…

Let $P_3$ denote the path on three vertices, and let $gr_k(K_3:n_1P_3, ldots,n_kP_3)$ be the corresponding multicolor Gallai-Ramsey number. For positive integers…

Let $G$ be a planar graph, and let an induced linear forest be a vertex-induced forest whose maximum degree is at most $2$. Chappel's conjecture. Every planar graph admits an induc…

Let $P_n$ be the path on $n$ vertices, let $F$ be a non-empty linear forest on $m$ vertices, let $K_1\vee F$ denote the join of $F$ with a single vertex, and let $o(F)$ be the numb…