105 research outputs found
Beyond Ohba's Conjecture: A bound on the choice number of -chromatic graphs with vertices
Let denote the choice number of a graph (also called "list
chromatic number" or "choosability" of ). Noel, Reed, and Wu proved the
conjecture of Ohba that when . We
extend this to a general upper bound: . Our result is sharp for
using Ohba's examples, and it improves the best-known
upper bound for .Comment: 14 page
Sum list coloring, the sum choice number, and sc-greedy graphs
Let G=(V,E) be a graph and let f be a function that assigns list sizes to the
vertices of G. It is said that G is f-choosable if for every assignment of
lists of colors to the vertices of G for which the list sizes agree with f,
there exists a proper coloring of G from the lists. The sum choice number is
the minimum of the sum of list sizes for f over all choosable functions f for
G. The sum choice number of a graph is always at most the sum |V|+|E|. When the
sum choice number of G is equal to this upper bound, G is said to be sc-greedy.
In this paper, we determine the sum choice number of all graphs on five
vertices, show that trees of cycles are sc-greedy, and present some new general
results about sum list coloring.Comment: 14 pages, 11 figure
Every triangle-free induced subgraph of the triangular lattice is -choosable
International audienceA graph is -choosable if for any color list of size associated with each vertex, one can choose a subset of colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer , every finite triangle-free induced subgraph of the triangular lattice is -choosable
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