Choosability of a weighted path and free-choosability of a cycle

TL;DR

This paper establishes conditions for coloring weighted paths and addresses free-choosability of cycles, advancing understanding of list coloring with weights in graph theory.

Contribution

It provides necessary and sufficient conditions for weighted path colorability and solves the free-choosability problem for cycles.

Findings

Characterization of weighted path colorability

Solution to free-choosability of cycles

Conditions for list assignments in weighted graphs

Abstract

A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex $v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be $(L,w)$-colorable for some list assignments $L$. Furthermore, we solve the problem of the free-choosability of a cycle.