Abstract
The partial list colouring conjecture due to Albertson, Grossman, and Haas (2000) states that for every $s$-choosable graph $G$ and every assignment of lists of size $t$, $1 \leq t \leq s$, to the vertices of $G$ there is an induced subgraph of $G$ on at least $\frac{t|V(G)|}{s}$ vertices which can be properly coloured from these lists. In this paper, we show that the partial list colouring conjecture holds true for certain classes of graphs like claw-free graphs, graphs with chromatic number at least $\frac{|V(G)|-1}{2}$, chordless graphs, and series-parallel graphs.
Highlights
Consider a simple, undirected, and finite graph G
We show that the answer to this question is not always by explicitly constructing an infinite family of 3-choosable graphs where a largest induced
2-choosable subgraph of each graph in the family is of size at most
Summary
Consider a simple, undirected, and finite graph G. L. The list chromatic number of G, denoted by χL(G), is the minimum positive integer k such that G is k-choosable. Consider a graph G on n vertices whose list chromatic number is s. The question is not to colour all the vertices but to colour as many vertices as one can They conjectured that, given a t-assignment, one can always find an induced subgraph of size at least can be properly list coloured. We give a more formal description of the conjecture below: Partial list colouring conjecture (Conjecture 1 in [1]): Consider an arbitrary graph G on n vertices whose list chromatic number is s. Let λLt(G) denote the size of a largest induced subgraph of G that is Lt-list colourable. In [4] it was shown that λt(G)
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