A note on the double-critical graph conjecture

A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lov\'asz conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$

Partitions and Edge Colourings of Multigraphs

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G)> ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S, T) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for line graphs of multigraphs.