48 problems

Total Coloring Conjecture. For every graph $G$,

Let $G$ be a simple graph, let $\Delta(G)$ denote its maximum degree, and let $\chi_T(G)$ denote its total chromatic number, the minimum number of colors in a total coloring of $G$…

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…

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…

Pilsniak–Wozniak's conjecture. For every graph $G$,

Distance-3 matching extension conjecture. If $H$ is a distance-3 matching, then every total-$(\Delta+2)$-coloring of $H$ extends to a total-$(\Delta+2)$-coloring of $G$.

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…

Fu's conjecture. For every graph $G$, $G$ has an equitable total $k$-coloring for each

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 $\chi_f”(G)$ denote the fractional total chromatic number. Reed's conjecture. For every $\epsilon>0$ and every maximum degree $\Delta$, there exists a girth $g$ such that every…

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

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 $G$ 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…

Behzad's Total Coloring Conjecture. Every graph with maximum degree $Delta$ admits a total coloring with at most

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…

Let $G$ be a graph with maximum degree $\Delta$, and let its total chromatic number be the least number of colors needed to color its vertices and edges so that adjacent or inciden…

AVD-total coloring conjecture. For every graph $G$,

Let $G$ be a graph, and let $\chi”(G)$ denote the minimum number of colors in a total coloring of $G$, where vertices and edges are colored so that adjacent or incident elements re…

Let $G$ be a finite, simple, undirected graph with maximum degree $\Delta(G)=3$. An adjacent vertex distinguishing (AVD) total coloring is a proper total coloring of $G$ such that…

Let $G$ be a simple graph, let $\Delta(G)$ be its maximum degree, and let $\chi_t(G)$ be its total chromatic number, the minimum number of colors in a proper coloring of the vertic…

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…

Neighbor full sum distinguishing total coloring conjecture. For every connected graph $G$ of order at least three,

For integers $m,n\geq 3$, let $C_m$ and $C_n$ be cycle graphs, and let $C_m\times C_n$ denote their direct product. A graph is Type 1 when its total chromatic number equals its max…