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

Linear Choosability of Sparse Graphs

We study the linear list chromatic number, denoted \lcl(G), of sparse graphs. The maximum average degree of a graph $G$, denoted \mad(G), is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $\Delta(G)$ satisfies \lcl(G)\ge \ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if \mad(G)<12/5 and $\Delta(G)\ge 3$, then \lcl(G)=\ceil{\Delta(G)/2}+1, and we give an infinite family of examples to show that this result is best possible; (2) if \mad(G)<3 and $\Delta(G)\ge 9$, then \lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to show that the bound on \mad(G) cannot be increased in general; (3) if $G$ is planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure

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

Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure