205 research outputs found
Covering a cubic graph with perfect matchings
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and
Fulkerson can be stated as follows: there exist five perfect matchings of G
such that each edge of G is contained in at least one of them. Here, we prove
that in each bridgeless cubic graph there exist five perfect matchings covering
a portion of the edges at least equal to 215/231 . By a generalization of this
result, we decrease the best known upper bound, expressed in terms of the size
of the graph, for the number of perfect matchings needed to cover the edge-set
of G.Comment: accepted for the publication in Discrete Mathematic
Covering cubic graphs with matchings of large size
Let m be a positive integer and let G be a cubic graph of order 2n. We
consider the problem of covering the edge-set of G with the minimum number of
matchings of size m. This number is called excessive [m]-index of G in
literature. The case m=n, that is a covering with perfect matchings, is known
to be strictly related to an outstanding conjecture of Berge and Fulkerson. In
this paper we study in some details the case m=n-1. We show how this parameter
can be large for cubic graphs with low connectivity and we furnish some
evidence that each cyclically 4-connected cubic graph of order 2n has excessive
[n-1]-index at most 4. Finally, we discuss the relation between excessive
[n-1]-index and some other graph parameters as oddness and circumference.Comment: 11 pages, 5 figure
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
How many matchings cover the nodes of a graph?
Given an undirected graph, are there matchings whose union covers all of
its nodes, that is, a matching--cover? A first, easy polynomial solution
from matroid union is possible, as already observed by Wang, Song and Yuan
(Mathematical Programming, 2014). However, it was not satisfactory neither from
the algorithmic viewpoint nor for proving graphic theorems, since the
corresponding matroid ignores the edges of the graph.
We prove here, simply and algorithmically: all nodes of a graph can be
covered with matchings if and only if for every stable set we have
. When , an exception occurs: this condition is not
enough to guarantee the existence of a matching--cover, that is, the
existence of a perfect matching, in this case Tutte's famous matching theorem
(J. London Math. Soc., 1947) provides the right `good' characterization. The
condition above then guarantees only that a perfect -matching exists, as
known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953).
Some results are then deduced as consequences with surprisingly simple
proofs, using only the level of difficulty of bipartite matchings. We give some
generalizations, as well as a solution for minimization if the edge-weights are
non-negative, while the edge-cardinality maximization of matching--covers
turns out to be already NP-hard.
We have arrived at this problem as the line graph special case of a model
arising for manufacturing integrated circuits with the technology called
`Directed Self Assembly'.Comment: 10 page
On disjoint matchings in cubic graphs
For and a cubic graph let denote the maximum number
of edges that can be covered by matchings. We show that and . Moreover, it turns out that
.Comment: 41 pages, 8 figures, minor chage
Berge - Fulkerson Conjecture And Mean Subtree Order
Let be a graph, and be the vertex set and edge set of , respectively. A perfect matching of is a set of edges, , such that each vertex in is incident with exactly one edge in . An -regular graph is said to be an -graph if for each odd set , where denotes the set of edges with precisely one end in . One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph has five perfect matchings such that each edge of is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every -graph has perfect matchings that covers each at least once. In the first part of this thesis, I will provide a lower bound to number of perfect matchings needed to cover the edge set of an -graph. I will also present some new conjectures that might shade a light towards the generalized Berge conjecture. In the second part, I will present a proof of a conjecture stating that there exists a pair of graphs and with , and such that mean subtree order of is smaller then mean subtree order of
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