266 research outputs found
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
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
Bounded degree graphs and hypergraphs with no full rainbow matchings
Given a multi-hypergraph that is edge-colored into color classes , a full rainbow matching is a matching of that contains
exactly one edge from each color class . One way to guarantee the
existence of a full rainbow matching is to have the size of each color class
be sufficiently large compared to the maximum degree of . In this
paper, we apply a simple iterative method to construct edge-colored
multi-hypergraphs with a given maximum degree, large color classes, and no full
rainbow matchings. First, for every and , we construct
edge-colored -uniform multi-hypergraphs with maximum degree such
that each color class has size and there is no full
rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and
Meshulam (2005) is best possible. Second, we construct properly edge-colored
multigraphs with no full rainbow matchings which disprove conjectures of
Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings
to list edge-colorings and prove that a color degree generalization of Galvin's
theorem (1995) does not hold
Strong edge-colorings for k-degenerate graphs
We prove that the strong chromatic index for each -degenerate graph with
maximum degree is at most
- …