论文标题
加强Hadwiger的猜想，价格为$ 4 $ - 和$ 5 $ -Chronic Graphs
Strengthening Hadwiger's conjecture for $4$- and $5$-chromatic graphs
论文作者
论文摘要
哈德威格（Hadwiger）著名的着色猜想指出，每$ t $ chromation图都包含$ k_t $ -Minor。 Holroyd [公牛。伦敦数学。 Soc。 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$.我们在第一个$ t = 4 $的第一个开放案例中证明了这个猜想。值得注意的是，我们的结果还直接暗示了Hadwiger的猜想的更强版本，$ 5 $ - 奇异图：每$ 5 $ - 奇异图都包含$ k_5 $ -minor，带有Singleton分支机构。实际上，在$ 5 $ vertex-Critical图中，我们可以指定Singleton分支集合为图形的任何顶点。
Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a $5$-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.
