199 research outputs found
Near-colorings: non-colorable graphs and NP-completeness
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned
into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of
V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus
on complexity aspects of such colorings when l=2,3. More precisely, we prove
that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3,
either every planar graph with girth at least g is (k,j)-colorable or it is
NP-complete to determine whether a planar graph with girth at least g is
(k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine
whether a planar graph that is either (0,0,0)-colorable or
non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit
non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar
graphs with girth 7
Avoidability of formulas with two variables
In combinatorics on words, a word over an alphabet is said to
avoid a pattern over an alphabet of variables if there is no
factor of such that where is a
non-erasing morphism. A pattern is said to be -avoidable if there exists
an infinite word over a -letter alphabet that avoids . We consider the
patterns such that at most two variables appear at least twice, or
equivalently, the formulas with at most two variables. For each such formula,
we determine whether it is -avoidable, and if it is -avoidable, we
determine whether it is avoided by exponentially many binary words
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