# Choosability with Separation of Cycles and Outerplanar Graphs

We consider the following list coloring with separation problem of graphs: Given a graph G and integers a,b, find the largest integer c such that for any list assignment L of G with |L(v)|≤ a for any vertex v and |L(u)∩ L(v)|≤ c for any edge uv of G, there exists an assignment φ of sets of integers to the vertices of G such that φ(u)⊂ L(u) and |φ(v)|=b for any vertex v and φ(u)∩φ(v)=∅ for any edge uv. Such a value of c is called the separation number of (G,a,b). We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth g≥ 5.

## Authors

• 2 publications
• 6 publications
• ### The intersection of two vertex coloring problems

A hole is an induced cycle with at least four vertices. A hole is even i...
04/17/2019 ∙ by Angele M. Foley, et al. ∙ 0

• ### On the m-eternal Domination Number of Cactus Graphs

Given a graph G, guards are placed on vertices of G. Then vertices are s...
07/18/2019 ∙ by Václav Blažej, et al. ∙ 0

• ### List-three-coloring P_t-free graphs with no induced 1-subdivision of K_1,s

Let s and t be positive integers. We use P_t to denote the path with t v...
06/04/2020 ∙ by Maria Chudnovsky, et al. ∙ 0

• ### Dichotomizing k-vertex-critical H-free graphs for H of order four

For k ≥ 3, we prove (i) there is a finite number of k-vertex-critical (P...
06/30/2020 ∙ by Ben Cameron, et al. ∙ 0

• ### Flexibility of Planar Graphs – Sharpening the Tools to Get Lists of Size Four

A graph where each vertex v has a list L(v) of available colors is L-col...
04/23/2020 ∙ by Ilkyoo Choi, et al. ∙ 0

• ### Least conflict choosability

Given a multigraph, suppose that each vertex is given a local assignment...
03/29/2018 ∙ by Zdeněk Dvořák, et al. ∙ 0