19 problems

A total graph is an ordered triple $(V_0,V_1,E)$, where $V_0$ and $V_1$ are disjoint sets of empty and full vertices, respectively, and $E$ is a set of edges on $V_0\cup V_1$. A to…

A locally irregular graph is a graph in which adjacent vertices have distinct degrees. Let $\mathfrak{T}$ be a family of specific graphs of maximum degree at most three. Baudon et…

Local Irregularity Conjecture. Every connected graph $G\notin\mathfrak{T'}$ satisfies

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…

A graph $G$ is locally irregular if adjacent vertices have distinct degrees. A locally irregular coloring is an edge coloring whose color classes induce locally irregular subgraphs…

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.

Locally irregular decomposition conjecture. Each connected graph $G$ which does not belong to $\mathfrak{T}'$ is decomposable to $3$ 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…

Let $\mathfrak{T}$ be the recursively defined family beginning with the triangle $K_3$, and let $\mathfrak{T'}$ consist of $\mathfrak{T}$ together with all odd-length paths and odd…

A graph $G=(V,E)$ can be decomposed into $k$ locally irregular subgraphs if its edge set can be partitioned as … so that each graph $G_i=(V,E_i)$ has distinct degrees at the endpoi…

A graph $G=(V,E)$ has no isolated edges if none of its connected components is a single edge. A weighting $cspanomega:E\to\{1,2,3\}$ is sum-distinguishing if the weighted degrees ……

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…

For a graph $G$, consider decompositions of its edges into subgraphs such that every component of every subgraph is regular or locally irregular. The regular-irregular chromatic in…

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

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.