102 research outputs found
A note on anti-coordination and social interactions
This note confirms a conjecture of [Bramoull\'{e}, Anti-coordination and
social interactions, Games and Economic Behavior, 58, 2007: 30-49]. The
problem, which we name the maximum independent cut problem, is a restricted
version of the MAX-CUT problem, requiring one side of the cut to be an
independent set. We show that the maximum independent cut problem does not
admit any polynomial time algorithm with approximation ratio better than
, where is the number of nodes, and arbitrarily
small, unless P=NP. For the rather special case where each node has a degree of
at most four, the problem is still MAXSNP-hard.Comment: 7 page
On a problem of Erd\H{o}s and Rothschild on edges in triangles
Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by
H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which
is contained in at least one triangle, must contain an edge that is in at least
H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether
for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all
sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed
C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best
possible in terms of the range for C, as it is known that every N-vertex graph
with more than (N^2)/4 edges contains an edge that is in at least N/6
triangles.Comment: 8 page
Linear-time Algorithms for Eliminating Claws in Graphs
Since many NP-complete graph problems have been shown polynomial-time
solvable when restricted to claw-free graphs, we study the problem of
determining the distance of a given graph to a claw-free graph, considering
vertex elimination as measure. CLAW-FREE VERTEX DELETION (CFVD) consists of
determining the minimum number of vertices to be removed from a graph such that
the resulting graph is claw-free. Although CFVD is NP-complete in general and
recognizing claw-free graphs is still a challenge, where the current best
algorithm for a graph has the same running time of the best algorithm for
matrix multiplication, we present linear-time algorithms for CFVD on weighted
block graphs and weighted graphs with bounded treewidth. Furthermore, we show
that this problem can be solved in linear time by a simpler algorithm on
forests, and we determine the exact values for full -ary trees. On the other
hand, we show that CLAW-FREE VERTEX DELETION is NP-complete even when the input
graph is a split graph. We also show that the problem is hard to approximate
within any constant factor better than , assuming the Unique Games
Conjecture.Comment: 20 page
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
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