108 research outputs found
Perfect Matchings in Claw-free Cubic Graphs
Lovasz and Plummer conjectured that there exists a fixed positive constant c
such that every cubic n-vertex graph with no cutedge has at least 2^(cn)
perfect matchings. Their conjecture has been verified for bipartite graphs by
Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every
claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect
matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure
Tangle-tree duality: in graphs, matroids and beyond
We apply a recent duality theorem for tangles in abstract separation systems
to derive tangle-type duality theorems for width-parameters in graphs and
matroids. We further derive a duality theorem for the existence of clusters in
large data sets.
Our applications to graphs include new, tangle-type, duality theorems for
tree-width, path-width, and tree-decompositions of small adhesion. Conversely,
we show that carving width is dual to edge-tangles. For matroids we obtain a
duality theorem for tree-width.
Our results can be used to derive short proofs of all the classical duality
theorems for width parameters in graph minor theory, such as path-width,
tree-width, branch-width and rank-width.Comment: arXiv admin note: text overlap with arXiv:1406.379
Tangle-tree duality in abstract separation systems
We prove a general width duality theorem for combinatorial structures with
well-defined notions of cohesion and separation. These might be graphs and
matroids, but can be much more general or quite different. The theorem asserts
a duality between the existence of high cohesiveness somewhere local and a
global overall tree structure.
We describe cohesive substructures in a unified way in the format of tangles:
as orientations of low-order separations satisfying certain consistency axioms.
These axioms can be expressed without reference to the underlying structure,
such as a graph or matroid, but just in terms of the poset of the separations
themselves. This makes it possible to identify tangles, and apply our
tangle-tree duality theorem, in very diverse settings.
Our result implies all the classical duality theorems for width parameters in
graph minor theory, such as path-width, tree-width, branch-width or rank-width.
It yields new, tangle-type, duality theorems for tree-width and path-width. It
implies the existence of width parameters dual to cohesive substructures such
as -blocks, edge-tangles, or given subsets of tangles, for which no width
duality theorems were previously known.
Abstract separation systems can be found also in structures quite unlike
graphs and matroids. For example, our theorem can be applied to image analysis
by capturing the regions of an image as tangles of separations defined as
natural partitions of its set of pixels. It can be applied in big data contexts
by capturing clusters as tangles. It can be applied in the social sciences,
e.g. by capturing as tangles the few typical mindsets of individuals found by a
survey. It could also be applied in pure mathematics, e.g. to separations of
compact manifolds.Comment: We have expanded Section 2 on terminology for better readability,
adding explanatory text, examples, and figures. This paper replaces the first
half of our earlier paper arXiv:1406.379
Number of cliques in graphs with a forbidden subdivision
We prove that for all positive integers , every -vertex graph with no
-subdivision has at most cliques. We also prove that
asymptotically, such graphs contain at most cliques, where
tends to zero as tends to infinity. This strongly answers a question
of D. Wood asking if the number of cliques in -vertex graphs with no
-minor is at most for some constant .Comment: 10 pages; to appear in SIAM J. Discrete Mat
Unifying duality theorems for width parameters in graphs and matroids. II. General duality
We prove a general duality theorem for tangle-like dense objects in
combinatorial structures such as graphs and matroids. This paper continues, and
assumes familiarity with, the theory developed in [6]Comment: 19 page
Rank connectivity and pivot-minors of graphs
The cut-rank of a set in a graph is the rank of the submatrix of the adjacency matrix over the binary field. A split is a
partition of the vertex set into two sets such that the cut-rank of
is less than and both and have at least two vertices. A graph is
prime (with respect to the split decomposition) if it is connected and has no
splits. A graph is -rank-connected if for every set of
vertices with the cut-rank less than , or is less than . We prove that every prime
-rank-connected graph with at least vertices has a prime
-rank-connected pivot-minor such that . As a corollary, we show that every excluded pivot-minor for the
class of graphs of rank-width at most has at most
vertices for . We also show that the excluded pivot-minors for the
class of graphs of rank-width at most have at most vertices.Comment: 19 pages; Lemma 5.3 is now fixe
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