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As a natural extension of the Four Color Theorem, Hajós conjectured that graphs containing no ‐subdivision are 4‐colorable. Any possible counterexample to this conjecture with minimum number of vertices is called a Hajós graph. Previous results show that Hajós graphs are 4‐connected but not 5‐connected. A ‐separation in a graph is a pair of edge‐disjoint subgraphs of such that...
It was conjectured by Hajós that graphs containing no ‐subdivision are 4‐colorable. Previous results show that any possible minimum counterexample to Hajós' conjecture, called Hajós graph, is 4‐connected but not 5‐connected. In this paper, we show that if a Hajós graph admits a 4‐cut or 5‐cut with a planar side then the planar side must be small or contains a special wheel. This is a step in our...
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