196 research outputs found
On the Parameterized Complexity of the Acyclic Matching Problem
A matching is a set of edges in a graph with no common endpoint. A matching M
is called acyclic if the induced subgraph on the endpoints of the edges in M is
acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for
an acyclic matching of size k in G. The problem is known to be NP-complete. In
this paper, we investigate the complexity of the problem in different aspects.
First, we prove that the problem remains NP-complete for the class of planar
bipartite graphs of maximum degree three and arbitrarily large girth. Also, the
problem remains NP-complete for the class of planar line graphs with maximum
degree four. Moreover, we study the parameterized complexity of the problem. In
particular, we prove that the problem is W[1]-hard on bipartite graphs with
respect to the parameter k. On the other hand, the problem is fixed parameter
tractable with respect to the parameters tw and (k, c4), where tw and c4 are
the treewidth and the number of cycles with length 4 of the input graph. We
also prove that the problem is fixed parameter tractable with respect to the
parameter k for the line graphs and every proper minor-closed class of graphs
(including planar graphs)
Rainbow Subgraphs in Edge-colored Complete Graphs -- Answering two Questions by Erd\H{o}s and Tuza
An edge-coloring of a complete graph with a set of colors is called
completely balanced if any vertex is incident to the same number of edges of
each color from . Erd\H{o}s and Tuza asked in whether for any graph
on edges and any completely balanced coloring of any sufficiently
large complete graph using colors contains a rainbow copy of . This
question was restated by Erd\H{o}s in his list of ``Some of my favourite
problems on cycles and colourings''. We answer this question in the negative
for most cliques by giving explicit constructions of respective
completely balanced colorings. Further, we answer a related question concerning
completely balanced colorings of complete graphs with more colors than the
number of edges in the graph .Comment: 8 page
- β¦