Talk:Vizing's theorem

Condition (1) in the proof is equivalent?

Suppose that no proper $(Δ+1)$ -edge-coloring of $G$ exists. This is equivalent to this statement:

(1) Let $xy \in E$ and $c$ be arbitrary proper $(Δ+1)$ -edge-coloring of $G - xy$ and $α$ be missing from $x$ and $β$ be missing from $y$ with respect to $c$ . Then the $α/β$ -path from $y$ ends in $x$ .

I am not entirely convinced that (1) is sufficient, i.e. (1) implies that no proper $(Δ+1)$ -edge-coloring of $G$ exists. Not entirely sure how you can restrict the coloring. What if $xy$ has some color $\gamma$ that is different from $\alpha$ and $\beta$ ?

I also checked the source and Diestel didn't mention anything about its sufficiency. DeyaoChen (talk) 01:20, 5 December 2024 (UTC)[reply]