12 problems

Let $G$ be a connected graph with at least three vertices. An edge weighting assigns a weight from $\{1,2,3\}$ to every edge of $G$, and the sum at a vertex is the sum of the weigh…

Let $B$ be the bow-tie cactus with $\mathrm{lir}(B)=4$. For a connected graph $G$ that is locally irregular colorable, let $\mathrm{lir}(G)$ denote the minimum number of colors in…

Local Irregularity Conjecture. If $G\neq B$, then

Four-subgraph conjecture. Each connected graph $G$ which does not belong to $\mathfrak{T}'$ is decomposable to $4$ locally irregular subgraphs.

Let $G$ be a graph, and let $\mathcal{M}^{[3]}(G)$ be the family of multigraphs obtained from $G$ by edge multiplication with edge multiplicities at most $3$. A multigraph is local…

Let $\mathfrak{T'}$ be the family consisting of the recursively defined family $\mathfrak{T}$, all odd-length paths and all odd-length cycles, and let $B$ denote the bow-tie graph.…

Let $\mathfrak{T'}$ be the family consisting of the recursively defined family $\mathfrak{T}$, all odd-length paths and all odd-length cycles. For a graph $G$, let ${\rm lir}(G)$ b…

A subgraph is called regular or locally irregular according as it is regular or locally irregular. The regular-irregular number $\operatorname{reg\text{-}irr}(G)$ is the minimum nu…

Let $G$ be a finite simple decomposable graph, meaning that it admits a locally irregular edge-coloring. Its locally irregular chromatic index, denoted by $\chi_{\mathrm{irr}}'(G)$…

Baudon–Bensmail–Przybyło–Woźniak conjecture. For every decomposable graph $G$,

Three-colour decomposition conjecture. Every connected graph $G$ which does not belong to $\mathfrak{T}$ and is neither an odd-length path nor an odd-length cycle can be decomposed…

1-2-3 Conjecture. Every graph $G$ containing no isolated edges admits a neighbour sum distinguishing $3$-edge colouring.