Journal Details
Format
Journal
eISSN
2083-5892
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English
Open Access

# Choosability with Separation of Cycles and Outerplanar Graphs

###### Accepted: 21 Feb 2021
Journal Details
Format
Journal
eISSN
2083-5892
First Published
13 Apr 2013
Publication timeframe
4 times per year
Languages
English

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.

#### MSC 2010

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• #### A Note About Monochromatic Components in Graphs of Large Minimum Degree

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