We study list coloring of common independent sets of two matroids. We construct a graphic matroid $M_1$ and a partition matroid $M_2$ with common chromatic number two and common list chromatic number three, showing that the two parameters need not be equal. This resolves a question raised by Kir\'aly, later stated as a conjecture by Aharoni, Berger, Guo, and Kotlar. We also show that if two strongly base-orderable matroids are each $2$-colorable, then their intersection is $2$-list-colorable.
A graph $H$ is called common if the number of monochromatic copies of $H$ in a $2$-edge-coloring of a large complete graph is asymptotically minimized by the random coloring with an equal proportion of each color. The house graph and $K_4$ with one edge subdivided once are known to be common by the flag algebra method....
We introduce and begin the study of sequence b-colorings, a natural generalization of the classical notion of b-colorings introduced by Irving and Manlove in 1999. In a sequence b-coloring, each color class is required to contain a prescribed minimum number of color-dominating vertices (CDVs). We establish several fund...
We exhibit a graph $G$ with $14$ vertices and list chromatic number equal to $3$ such that there is a $2$-list assignment $L$ of $G$ such that at most $9$ vertices of $G$ can be properly coloured from $L$. This disproves the Partial List Colouring Conjecture of Albertson, Grossman and Haas. This counterexample was disc...
In 1971, Mason conjectured that the numbers of independent sets of fixed size in a matroid constitute an ultra log-concave sequence. In 2020, this conjecture was proven by Br\"and\'en and Huh and independently by Anari, Liu, Gharan and Vinzant. Recently, this result was extended to $M^\natural$-concave functions. In th...
We extend the conjecture of Erd\H{o}s and Gy\'arf\'as on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, th...
We exhibit a cubic graph with total chromatic number equal to four and list total chromatic number five. This disproves the List Total Colouring Conjecture of Borodin--Kostochka--Woodall, Juvan--Mohar--\v{S}krekovski and Hilton--Johnson from the late 1990s. ChatGPT 6 Astra Ultra discovered the counterexample with littl...
Jonathan A. Noel· 0 citations
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