Chromatic λ‐choosable and λ‐paintable graphs

Abstract

Let ϕ ( k ) be the minimum number of vertices in a non‐ k‐choosable k‐chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that ϕ ( k ) ≥ 2 k + 2. This bound is tight if k is even. If k is odd, then it is known that ϕ ( k ) ≤ 2 k + 3 and it is conjectured by Noel that ϕ ( k ) = 2 k + 3. For a multiset λ = { k 1 , k 2 , … , k q } of positive integers, let k λ = ∑ i = 1 q k i. For positive integer a, let m λ ( a ) be the multiplicity of a in λ, and let m λ ( odd ) be the number of elements in λ that are odd integers. A λ‐list assignment of G is a list assignment L such that the colour set ∪ v ∈ V ( G ) L ( v ) can be partitioned into the disjoint union C 1 ∪ C 2 ∪ … ∪ C q of q sets so that for each i and each vertex v of G, ∣ L ( v ) ∩ C i ∣ ≥ k i. We say G is λ‐choosable if G is L‐colourable for any λ‐list assignment L of G. We say λ is trivial if λ consists of k λ copies of 1. It is easy to see that if λ is trivial, then λ‐choosable is equivalent to k λ‐colourable. For any nontrivial λ, let ϕ ( λ ) be the minimum number of vertices in a non‐ λ‐choosable k λ‐chromatic graph. We prove that for any nontrivial λ, 2 k λ + m λ ( 1 ) + 2 ⩽ ϕ ( λ ) ⩽ 2 k λ + m λ ( odd ) + 2. In particular, if m λ ( 1 ) = m λ ( odd ) = t, that is, λ contains no odd integer greater than 1, then ϕ ( λ ) = 2 k λ + t + 2. We also prove that ϕ ( λ ) ⩽ 2 k λ + 5 m λ ( 1 ) + 3. In particular, if m λ ( 1 ) = 0, then 2 k λ + 2 ⩽ ϕ ( λ ) ⩽ 2 k λ + 3. We then introduce the concept of λ‐paintability of graphs, which is an online version of λ‐choosability. Let ψ ( λ ) be the minimum number of vertices in a non‐ λ‐paintable k λ‐chromatic graph. We determine the value of ψ ( λ ) when each integer in λ is at most 2.

Bad list assignments for non‐k $k$‐choosable k $k$‐chromatic graphs with 2k+2 $2k+2$‐vertices

“…Noel [6] conjectured that if χ G ( ) is odd, then the bound is not tight, that is, if k is odd, then all k-chromatic graphs with k 2 + 2 vertices are k-choosable. This conjecture was confirmed in [12], where the following result was proved.…”

ZDP(n) ${Z}_{DP}(n)$ is bounded above by n2−(n+3)∕2 ${n}^{2}-(n+3)\unicode{x02215}2$

“…Note that the condition $$\chi (G)\ge (| V(G)| -1)\unicode{x02215}2$$ in Theorem 2 is the best possible due to many infinite classes of graphs. In addition, Zhu and Zhu [11] have characterized all the complete multipartite graphs $$G$$ with $$\chi (G)=(| V(G)| -2)\unicode{x02215}2$$ and $$\chi (G)\lt {\chi }_{\ell }(G)$$. More relevant results can also be found in [10].…”