Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, April 12, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Raphael Steiner

New bounds for relatives of Hadwiger's conjecture

In this talk, I will present some recent results on two variants of Hadwiger's conjecture. First, I will discuss the so-called Odd Hadwiger's conjecture, a strengthening of Hadwiger's conjecture proposed by Gerards and Seymour in 1995. First I will show a new upper bound for coloring graphs with no odd K_t-minor, thereby improving previous bounds for this problem, which comes along with a substantially simpler proof. Second, I will show a new bound of 2t-2 for clustered colorings of odd K_t-minor free graphs, in which we look for (possibly improper) colorings with monochromatic components of small size, this also significantly improves previous results. Third, I will discuss the so-called List Hadwiger's conjecture, which states that there exists a constant c such that every graph with no Kt-minor is ct-choosable (i.e., list colorable). I will present a new lower bound 2t-o(t) for list coloring K_t-minor free graphs which refutes a conjecture by Kawarabayashi and Mohar. Fourth, I will present a result saying that K_{s,t}-minor free graphs for large comparable s and t can have list chromatic number at least (1-o(1)(s+t+min(s,t)), this refutes a conjecture by Woodall.

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