287 research outputs found
Vertex coloring of plane graphs with nonrepetitive boundary paths
A sequence is a repetition. A sequence
is nonrepetitive, if no subsequence of consecutive terms of form a
repetition. Let be a vertex colored graph. A path of is nonrepetitive,
if the sequence of colors on its vertices is nonrepetitive. If is a plane
graph, then a facial nonrepetitive vertex coloring of is a vertex coloring
such that any facial path is nonrepetitive. Let denote the minimum
number of colors of a facial nonrepetitive vertex coloring of . Jendro\vl
and Harant posed a conjecture that can be bounded from above by a
constant. We prove that for any plane graph
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
Rainbow matchings in bipartite multigraphs
Suppose that is a non-negative integer and a bipartite multigraph is
the union of matchings
, each of size . We show that has a rainbow matching of
size , i.e. a matching of size with all edges coming from different
's. Several choices of parameters relate to known results and conjectures
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