Choice number of some complete multi-partite graphs

List Colouring When The Chromatic Number Is Close To the Order Of The Graph

“…In his paper Ohba proved this conjecture for graphs G with |V (G)| ≤ χ(G) + 2χ(G). The conjecture was settled for some other special cases in [2]. Nevertheless, until the present there was no result which shows that χ(G) = χ l (G) for all graph with |V (G)| ≤ αχ(G) with some α strictly bigger than 1.…”

On chromatic‐choosable graphs

“…We propose the following conjecture. The assertion of the conjecture is best possible if it is true, because it is shown in [1] that there exist several graphs G such that jVðGÞj ¼ 2ðGÞ þ 2 and chðGÞ > ðGÞ. For example, let G be the complete k-partite graph with one partite set of order 4 and k À 1 partite sets of order 2.…”

Ohba’s conjecture for graphs with independence number five

“…Part (i) of Theorem C follows from Theorem B(iii) if t = 0, and it was proved by Enomoto et al [1] for t ⩾ 1; for t ⩾ 2 it follows from Theorem B(vii). Part (ii) of Theorem C follows from part (i) if t = 0 and from Theorem B(iv) if t = 1; it was proved by Shen et al [11] for t = 2, 3, and by Shen et al [12] for t = 4.…”

“…Thus R contains all σ (singleton) subsets of V p , and L(x) ∪ L(y) = C Q for each pair of distinct elements x, y ∈ V p . By (1), this means that each color in C Q is in the list of exactly σ − 1 vertices of V p , so that |C Q | = σ k/(σ − 1), and for each vertex x ∈ V p , L(x) omits a different set of k/(σ − 1) colors from C Q . Thus, for each two distinct vertices x, y ∈ V p ,…”