23 problems

For a multigraph $G$, a total coloring colors vertices and edges so that adjacent vertices, incident edges, and incident vertex-edge pairs receive distinct colors. Let $\chi”(G)$ b…

Let $G$ be a graph. A total coloring is a function $g:V\cup E\to\{1,2,\dots,k\}$, with total vertex weight $\sigma^t(v)=g(v)+\sum_{v\in e}g(e)$. Adjacent vertices are distinguished…

Total-coloring extension conjecture for matchings. If $H$ is a matching, then every total-$(\Delta+3)$-coloring of $H$ in $G$ extends to a total-$(\Delta+3)$-coloring of $G$.

Let $G$ be a finite, simple, undirected graph. For each vertex $u$, let $N_G(u)$ be its set of neighbours, let $\partial_G(u)$ be the set of edges incident with $u$, and let…

For a graph $G$, let $\Delta(G)$ denote its maximum degree, and let $\chi\,”_e(G)$ denote its equitable total chromatic number, namely the least number of colors in an equitable to…

Let $G$ be a simple graph. The graph $G^{\frac{2}{2}}$ is the square of the subdivision of $G$ obtained by replacing every edge by a path of length two, and $\chi$ and $\omega$ den…

Let $G$ be a nice graph and let $w:V(G)\cup E(G)\rightarrow\{1,2,\ldots,k\}$ be a total integer weighting. For each vertex $v$, define … where $N(v)$ is the open neighborhood of…

Goldberg's conjecture. If

Campos–de Mello's conjecture. If $G=C_n^k$, then

Let $G$ be a graph with at least two vertices. The neighbor product distinguishing total chromatic number $\chi”_{\prod}(G)$ is the smallest number of colors in a neighbor product…

Let $G$ be a graph and let $T(G)$ be its total graph, whose vertices represent the vertices and edges of $G$. An equitable $k$-coloring is a proper $k$-coloring whose color classes…

Let $G$ be a simple graph. Write $\chi'(G)$ and $\chi”(G)$ for its edge-chromatic and total-chromatic numbers, and $\chi'_{\ell}(G)$ and $\chi”_{\ell}(G)$ for the corresponding lis…

Let $G$ be a finite, simple, undirected graph, let $p$ be a positive integer, and let $\lambda^T_p(G)$ denote the minimum $k$ for which $G$ has a $(p,1)$-total $k$-labelling. Havet…

Let $K_n$ be the complete graph on $n$ vertices, let $L(K_n)$ be its line graph, and let $\chi”$ denote total chromatic number. Vignesh et al.'s conjecture. For every complete grap…

Let $G$ be a 3-clique graph, and let Hilton's condition mean the condition for a simple graph with an even number of vertices and a universal vertex that … where…

Linear vertex-distinguishing total coloring conjecture. For some positive constant $c$,

Vertex-distinguishing total coloring conjecture.

Let $G$ be a finite multigraph, and let $T(G)$ be its total graph. Write $\chi(T(G))$ for the chromatic number and $\chi_l(T(G))$ for the list chromatic number. Borodin–Kostochka–W…

Let $G$ be a simple graph and let $H$ be a proper subgraph of $G$. Write $\chi \, ”_{(8)}(G)$ for the minimum number of colors in an $(8)$-distinguishing total coloring. Monotonici…

Equitable total coloring conjecture. For every graph $G$,

Let $\Gamma$ be a $k$-regular graph. The graph is called $tndi_{\sum}$ class II when its adjacent vertex distinguishing index by sum attains the class-II value. Class II characteri…

For a graph $\Gamma$, let $tndi_{\sum}(\Gamma)$ be the minimum number of colors in a proper total coloring distinguishing adjacent vertices by sums. The graph is called…

Total 1-2 Conjecture. For every graph $G$,