508 research outputs found
Some Conclusion on Unique k
If a graph G admits a k-list assignment L such that G has a unique L-coloring, then G is called uniquely k-list colorable graph, or UkLC graph for short. In the process of characterizing UkLC graphs, the complete multipartite graphs K1*r,s(r,sβN) are often researched. But it is usually not easy to construct the unique k-list assignment of K1*r,s. In this paper, we give some propositions about the property of the graph K1*r,s when it is UkLC, which provide a very significant guide for constructing such list assignment. Then a special example of UkLC graphs K1*r,s as a application of these propositions is introduced. The conclusion will pave the way to characterize UkLC complete multipartite graphs
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
Independent transversals in locally sparse graphs
Let G be a graph with maximum degree \Delta whose vertex set is partitioned
into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G)
containing exactly one vertex from each part V_i. If it is also an independent
set, then we call it an independent transversal. The local degree of G is the
maximum number of neighbors of a vertex v in a part V_i, taken over all choices
of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all
part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta),
then G has an independent transversal for sufficiently large \Delta. This
extends several previous results and settles (in a stronger form) a conjecture
of Aharoni and Holzman. We then generalize this result to transversals that
induce no cliques of size s. (Note that independent transversals correspond to
s=2.) In that context, we prove that parts of size |V_i| >=
(1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence
of such a transversal, and we provide a construction that shows this is
asymptotically tight.Comment: 16 page
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