Light subgraphs in graphs with average degree at most four

A graph $H$ is said to be {\em light} in a family $\mathfrak{G}$ of graphs if at least one member of $\mathfrak{G}$ contains a copy of $H$ and there exists an integer $\lambda(H, \mathfrak{G})$ such that each member $G$ of $\mathfrak{G}$ with a copy of $H$ also has a copy $K$ of $H$ such that $\deg_{G}(v) \leq \lambda(H, \mathfrak{G})$ for all $v \in V(K)$. In this paper, we study the light graphs in the class of graphs with small average degree, including the plane graphs with some restrictions on girth.Comment: 12 pages, 18 figure

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