Abstract
Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a Kt-minor. Holroyd[11]conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G, then G contains a Kt-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 K5-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.
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