Minimal counterexamples and discharging method

Recently, the author found that there is a common mistake in some papers by using minimal counterexample and discharging method. We first discuss how the mistake is generated, and give a method to fix the mistake. As an illustration, we consider total coloring of planar or toroidal graphs, and show that: if $G$ is a planar or toroidal graph with maximum degree at most $\kappa - 1$, where $\kappa \geq 11$, then the total chromatic number is at most $\kappa$.Comment: 8 pages. Preliminary version, comments are welcom

Covering line graphs with equivalence relations

An equivalence graph is a disjoint union of cliques, and the equivalence number $\mathit{eq}(G)$ of a graph $G$ is the minimum number of equivalence subgraphs needed to cover the edges of $G$. We consider the equivalence number of a line graph, giving improved upper and lower bounds: $\frac 13 \log_2\log_2 \chi(G) < \mathit{eq}(L(G)) \leq 2\log_2\log_2 \chi(G) + 2$. This disproves a recent conjecture that $\mathit{eq}(L(G))$ is at most three for triangle-free $G$; indeed it can be arbitrarily large. To bound $\mathit{eq}(L(G))$ we bound the closely-related invariant $\sigma(G)$, which is the minimum number of orientations of $G$ such that for any two edges $e,f$ incident to some vertex $v$, both $e$ and $f$ are oriented out of $v$ in some orientation. When $G$ is triangle-free, $\sigma(G)=\mathit{eq}(L(G))$. We prove that even when $G$ is triangle-free, it is NP-complete to decide whether or not $\sigma(G)\leq 3$.Comment: 10 pages, submitted in July 200

On edge-group choosability of graphs

In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our claim are graphs with maximum degree less than 4, planar graphs with maximum degree at least 11, planar graphs without small cycles, outerplanar graphs and near-outerplanar graphs