3,848 research outputs found
The chromatic numbers of double coverings of a graph
If we fix a spanning subgraph of a graph , we can define a chromatic
number of with respect to and we show that it coincides with the
chromatic number of a double covering of with co-support . We also find
a few estimations for the chromatic numbers of with respect to .Comment: 10 page
Decompositions into subgraphs of small diameter
We investigate decompositions of a graph into a small number of low diameter
subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E)
on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that
|E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the
subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma,
Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a
constant depending only \epsilon. This shows that every dense graph can be
partitioned into a small number of ``small worlds'' provided that few edges can
be ignored. Improving on their result, we determine P(n,\epsilon,d) within an
absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded
for \epsilon
n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We
also prove that if G has large minimum degree, all the edges of G can be
covered by a small number of low diameter subgraphs. Finally, we extend some of
these results to hypergraphs, improving earlier work of Polcyn, R\"odl,
Ruci\'nski, and Szemer\'edi.Comment: 18 page
Covering line graphs with equivalence relations
An equivalence graph is a disjoint union of cliques, and the equivalence
number of a graph is the minimum number of equivalence
subgraphs needed to cover the edges of . We consider the equivalence number
of a line graph, giving improved upper and lower bounds: . This disproves a
recent conjecture that is at most three for triangle-free
; indeed it can be arbitrarily large.
To bound we bound the closely-related invariant
, which is the minimum number of orientations of such that for
any two edges incident to some vertex , both and are oriented
out of in some orientation. When is triangle-free,
. We prove that even when is triangle-free, it
is NP-complete to decide whether or not .Comment: 10 pages, submitted in July 200
Matchings, coverings, and Castelnuovo-Mumford regularity
We show that the co-chordal cover number of a graph G gives an upper bound
for the Castelnuovo-Mumford regularity of the associated edge ideal. Several
known combinatorial upper bounds of regularity for edge ideals are then easy
consequences of covering results from graph theory, and we derive new upper
bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio
Problems and memories
I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the
centennial conference in Budapest to honor Paul Erd\H{o}
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
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