8 problems

Let $P_k$ be the path on $k$ vertices, let $K_s$ be the complete graph on $s$ vertices, and let $sat(n,H)$ denote the minimum number of edges in an $H$-saturated graph on $n$ verti…

Let $P_n$ be the path on $n$ vertices, let $mP_n$ denote the disjoint union of $m$ copies of $P_n$, let $K_1$ be a single vertex, and let $\chi_{la}(G)$ denote the local antimagic…

Full-degree vertex necessity conjecture. If

Local antimagic chromatic number conjecture.

Let $G$ be a simple graph. Write $\chi=\chi(G)$ for its chromatic number and $\Delta=\Delta(G)$ for its maximum degree. For graphs $G_1$ and $G_2$, let $G_1\lor G_2$ denote their j…

Let $G$ and $H$ be graphs such that $q(G)=q(H)=2$, with $MB(G)=k_1$ and $MB(H)=k_2$, and suppose that the quantities denoted in the source by $\mrr+(G)$ and $\mrr+(H)$ satisfy…

Let $G$ and $H$ be two graphs with $q(G)=q(H)=2$ and $MB(G)=MB(H)$. Join multiplicity-bipartition conjecture. One should have $MB(G\vee H)=MB(G)$, and this should hold in a more ge…

Let $P_m$ and $P_n$ be paths on $m+1$ and $n+1$ vertices, respectively, and let $P_m+P_n$ denote their graph join. The minimum coprime number $\mathfrak{pr}(G)$ is the least $M$ fo…