17 problems

A packing $(1,1,2,3)$-coloring is a partition of the vertex set into four classes whose pairwise vertex distances are respectively at least $2$, $2$, $3$, and $4$. Gastineau–Togni…

Let $G$ be a $(3,0)$-saturated subcubic graph, and let $g_3(G)$ denote its local girth parameter. Packing coloring conjecture. If … then $G$ is $(1,1,2)$-packing colorable. The pap…

For a graph $G$, its 1-subdivision is obtained by replacing every edge with a path of two edges. The packing chromatic number $\chi_p(G)$, or PCN, is the smallest positive integer…

Let $G$ be a claw-free subcubic graph. Seven-color packing coloring conjecture. The graph $G$ is $(2,2,2,2,2,2,2)$-packing colorable. The paper gives related packing-coloring resul…

Let $G$ be a $2$-saturated subcubic graph. Five-color packing coloring conjecture. The graph $G$ is $(1,2,2,2,2)$-packing colorable. The paper proves a four-color variant when…

Let $G$ be a $0$-saturated subcubic graph, and let $g_3(G)$ denote its local girth parameter. Packing chromatic number conjecture. If … then … The paper proves the corresponding up…

A packing $(1,1,2,2)$-coloring of a graph is a partition of its vertex set into four classes whose pairwise vertex distances are respectively at least $2$, $2$, $3$, and $3$. Liu–Z…

Let $G$ be a claw-free subcubic graph, and let $\mathcal{G}_{11}$ be the graph specified in the paper. Exceptional packing coloring conjecture. If $G\neq\mathcal{G}_{11}$, then $G$…

Let $G$ be a subcubic graph, meaning that every vertex of $G$ has degree at most $3$, and let $G'$ be a subdivision of $G$. The packing chromatic number conjecture. The packing chr…

Let $G$ be a graph in which every vertex has degree at most $3$, and let $S(G)$ be the graph obtained by subdividing every edge of $G$, replacing each edge $uv$ by a path $u$-$w$-…

Let $P_{\ell t}\Diamond_{\ell}K_6$ denote the path-aligned product of a path and the complete graph $K_6$, and let $\chi_p(G)$ be the packing chromatic number of a graph $G$. Twent…

Let $P_\infty$ be the two-way infinite path with vertex set $\mathbb{Z}$, and let $P_\infty \boxtimes P_\infty$ be the infinite diagonal grid with vertex set…

Let $k\geq 3$, and let a $k$-degree graph mean a graph of maximum degree at most $k$. A coloring is $(1^{k-1},2^{k-1},3)$-packing colorable if its vertices can be partitioned into…

Let $G$ be a subcubic graph, and let $D(G)$ denote its subdivision, obtained by replacing each edge of $G$ by a path of length two. The packing chromatic number $\chi_p(G)$ is the…

Let $G$ be a subcubic graph, meaning that its maximum degree is at most $3$. Let $D(G)$ be the graph obtained by subdividing every edge of $G$, and let $\chi_p(G)$ denote the packi…

Let $G$ be a graph, and write $\a0\omega(G)$ for its clique number, $\a0\chi(G)$ for its chromatic number, and $\a0\chi_{\rho}(G)$ for its packing chromatic number. Non-realizabili…

Packing-coloring conjecture. There exist oriented generalized theta graphs with packing chromatic number $5$ containing no path of length less than $4$.