European Journal of Combinatorics最新文献

We consider Kempe changes on the $k$-colorings of a graph on $n$ vertices. If the graph is $(k-1)$-degenerate, then all its $k$-colorings are equivalent up to Kempe changes. However, the sequence between two $k$-colorings that arises from the proof may have length exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most $k-1$. Namely, any two $k$-colorings are equivalent up to $O\left(k{n}^2\right)$ Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As one of the main results, we derive that given an $n$-vertex graph with maximum degree $\Delta$, the $\Delta$-colorings are all equivalent up to $O_{\Delta }\left(n^2\right)$ Kempe changes, unless $\Delta =3$ and some connected component is a 3-prism, that is $K_2\square K_3$, in which case there exist some non-equivalent 3-colorings.

Let $r\ge 1$ be any non negative integer and let $G=(V,E)$ be any undirected graph in which a subset $D\subseteq V$ of vertices are initially infected. We consider the process in which, at every step, each non-infected vertex with at least $r$ infected neighbours becomes infected and an infected vertex never becomes non-infected. The problem consists in determining the minimum size $s_r\left(G\right)$ of an initially infected vertices set $D$ that eventually infects the whole graph $G$. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that $s_1\left(G\right)=1$ for any connected graph $G$. The case when $G$ is the $n\times n$ grid, $G_{n\times n}$, and $r=2$ is well known and appears in many puzzle books, in particular due to the elegant proof that shows that $s_2\left(G_{n\times n}\right)=n$ for all $n\in N$. We study the cases of square grids, $G_{n\times n}$, and tori, $T_{n\times n}$, when $r\in \{3,4\}$. We show that $s_3\left(G_{n\times n}\right)=\lceil \frac{n^2+2n+4}{3}\rceil$ for every $n$ even and that