13 problems

Let $r$ and $s$ be integers, with $r$ even and $r\ge 2$, and $s$ odd and $s\ge 3$. Let $rFB(s)$ denote the corresponding flower-bouquet graph, and let $\chi_{la}(G)$ be the local a…

Local antimagic chromatic number conjecture.

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…

Let $G$ and $H$ be graphs, let $G[H]$ be their lexicographic product, and let $\chi_{la}$ and $\chi$ denote the local antimagic chromatic number and chromatic number, respectively.…

Let $G$ and $H$ be disjoint non-null graphs. Write $G[H]$ for their lexicographic product, and let $\chi_{la}(G)$ denote the local antimagic chromatic number and $\chi(G)$ the chro…

Let $T$ be a tree with $k$ pendant vertices. The families $K(2;a,b)$ and $Ct(3;a,0,b)$ are the exceptional tree families described in the supplied statement. Almost-all-trees conje…

Let $G$ be a graph of order $n$, and let $K_n$, $K_{1,n-1}$, $K(m;n_1,n_2,\ldots,n_m)$, and $Ct(3;n_1,0,n_3)$ denote the graph families used in the paper. Order-equality classifica…

Let $G$ be a graph with $k\ge 1$ pendant vertices and chromatic number $\chi(G)$ satisfying $k\ge \chi(G)-1$. The graphs $K(2;a,b)$, caterpillar graphs in the cited theorem,…

Let $T_k$ be a tree with $k\ge 2$ pendant vertices. Caterpillars, spiders, and lobsters are standard classes of trees, and $\chi_{la}(T_k)$ denotes the local antimagic chromatic nu…

M{"o}bius ladder conjecture. For even $n\ge 4$,

Complete-join characterization. For $n\ge 2$,

Join characterization. For $n\ge 1$,

Let $GB(a_1,a_2,\ldots,a_r)$ denote the generalized book graph obtained by edge-gluing cycles of orders $a_1,a_2,\ldots,a_r$, where $a_1\ge a_2\ge \cdots\ge a_r\ge 3$ and $r\ge 2$.…