
The intersection of two vertex coloring problems
A hole is an induced cycle with at least four vertices. A hole is even i...
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On the meternal Domination Number of Cactus Graphs
Given a graph G, guards are placed on vertices of G. Then vertices are s...
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Listthreecoloring P_tfree graphs with no induced 1subdivision of K_1,s
Let s and t be positive integers. We use P_t to denote the path with t v...
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Dichotomizing kvertexcritical Hfree graphs for H of order four
For k ≥ 3, we prove (i) there is a finite number of kvertexcritical (P...
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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 Lcol...
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Least conflict choosability
Given a multigraph, suppose that each vertex is given a local assignment...
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2× n Grids have Unbounded AnagramFree Chromatic Number
We show that anagramfree vertex colouring a 2× n square grid requires a...
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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 freeseparation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and freeseparation number of the cycle and derive from them the freeseparation number of a cactus. We also present a lower bound for the separation and freeseparation numbers of outerplanar graphs of girth g≥ 5.
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