On the choice number of complete multipartite graphs with part size four

Abstract

a b s t r a c tLet ch(G) denote the choice number of a graph G, and let K s * k be the complete k-partite graph with s vertices in each part. Erdős, Rubin, and Taylor showed that ch(K 2 * k ) = k, and suggested the problem of determining the choice number of K s * k . The first author establishedHere we prove ch(K 4 * k ) =  3k−1 2  .

Beyond Ohba’s Conjecture: A bound on the choice number ofk-chromatic graphs withnvertices

“…v∈V (G) L(v) < n is a standard reduction for minimal counterexamples in choosability problems, so much so that it has a name: the "Small Pot Lemma". It has been applied in diverse situations, including [3,4,5,6,16,12].…”