Graphs with no $K_9^=$ minor are 10-colorable

Abstract

Hadwiger's conjecture claims that any graph with no K t minor is (t − 1)-colorable. This has been proved for t ≤ 6, but remains open for t ≥ 7. As a variant of this conjecture, graphs with no K = t minor have been considered, where K = t denotes the complete graph with two edges removed. It has been shown that graphs with no K = t minor are (2t − 8)-colorable for t ∈ {7, 8} [6,12]. In this paper, we extend this result to the case t = 9 and show that graphs with no K = 9 minor are 10-colorable.

A note on Hadwiger’s conjecture for W5-free graphs with independence number two

“…Rolek and Song [9] showed in 2017 that χ(G) ≤ 8, 9 and 12 for every K = 8 , K − 8 and K 9 minor-free graph, respectively. Rolek [8] showed later in 2018 that χ(G) ≤ 10 for every K = 9 minor-free graph. In this note we pay particular attention to Conjecture 1.1 for graphs G with α(G) ≤ 2.…”