On the choosability of claw-free perfect graphs

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most $4$

On the Coloring of Pseudoknots

Pseudodiagrams are diagrams of knots where some information about which strand goes over/under at certain crossings may be missing. Pseudoknots are equivalence classes of pseudodiagrams, with equivalence defined by a class of Reidemeister-type moves. In this paper, we introduce two natural extensions of classical knot colorability to this broader class of knot-like objects. We use these definitions to define the determinant of a pseudoknot (i.e. the pseudodeterminant) that agrees with the classical determinant for classical knots. Moreover, we extend Conway notation to pseudoknots to facilitate the investigation of families of pseudoknots and links. The general formulae for pseudodeterminants of pseudoknot families may then be used as a criterion for p-colorability of pseudoknots.Comment: 22 pages, 24 figure