12 problems

Let $G$ be a graph with signature $s(G)$, where the signature is the difference between the numbers of positive and negative eigenvalues of its adjacency matrix. Let $c_3(G)$ and…

Let $G$ be a 4-connected planar graph on $n$ vertices. A cycle of length $4$ is a cycle with four vertices. Malkevitch's conjecture. If $G$ contains a cycle of length $4$, then $G$…

Let $G[X,Y]$ be a balanced bipartite graph of order $2n$, let $S$ be a subset of $X$ with $|S|\geq 2k+2$, and let $S_{C_i}$ denote the vertices of $S$ on a cycle $C_i$. Prescribed…

Let $G[X,Y]$ be a balanced bipartite graph of order $2n$, let $S$ be a subset of $X$ with $|S|\geq 2k+1$, and let $\sigma_{1,1}(S)$ denote the degree-sum parameter used in the pape…

Weakly supercritical circumference conjecture.

Linear circumference conjecture.

Let $H$ and $H'$ be graphs with $m_2(H)\geq m_2(H')$. Asymmetric Ramsey-density conjecture. If neither $H$ nor $H'$ is a matching and at least one of $H$ or $H'$ contains a cycle,…

Let $k>1$, and let $C_{2k+1}$ denote the cycle of length $2k+1$. A graph is $C_{2k+1}$-free if it contains no subgraph isomorphic to $C_{2k+1}$; $K_{a,b}$ denotes the complete bipa…

Let $n\geq 4$. A triangle-free graph is a graph containing no triangle, and $K_{a,b}$ denotes the complete bipartite graph with parts of sizes $a$ and $b$. Durocher–Gunderson–Li–Sk…

Let $G$ be a graph, let $\overline{G}$ denote its complement, and for $k\geq 3$ let $f_k(G)$ be the number of frustrated $k$-cycles, where a cyclic ordering $v_1v_2\dots v_k$ is fr…

Let $T$ be a tournament on $n$ vertices, and let $c_3$ and $c_4$ denote its normalized numbers of cyclic triples and cyclic $4$-cycles, respectively. For a transitive tournament…

Let a cycle-maximal triangle-free graph be a triangle-free graph to which no edge can be added without creating a triangle. For a graph on $n$ vertices, let…