14,718 research outputs found
Counting edge-injective homomorphisms and matchings on restricted graph classes
We consider the -hard problem of counting all matchings with
exactly edges in a given input graph ; we prove that it remains
-hard on graphs that are line graphs or bipartite graphs
with degree on one side. In our proofs, we use that -matchings in line
graphs can be equivalently viewed as edge-injective homomorphisms from the
disjoint union of length- paths into (arbitrary) host graphs. Here, a
homomorphism from to is edge-injective if it maps any two distinct
edges of to distinct edges in . We show that edge-injective
homomorphisms from a pattern graph can be counted in polynomial time if
has bounded vertex-cover number after removing isolated edges. For hereditary
classes of pattern graphs, we complement this result: If the
graphs in have unbounded vertex-cover number even after deleting
isolated edges, then counting edge-injective homomorphisms with patterns from
is -hard. Our proofs rely on an edge-colored
variant of Holant problems and a delicate interpolation argument; both may be
of independent interest.Comment: 35 pages, 9 figure
Applications of entropy to extremal problems
The Sidorenko conjecture gives a lower bound on the number of homomorphisms from a bipartite graph to another graph. Szegedy [28] used entropy methods to prove the conjecture in some cases. We will refine these methods to also give lower bounds for the number of injective homomorphisms from a bipartite graph to another bipartite graph, and a lower bound for the number of homomorphisms from a k-partite hypergraph to another k-partite hypergraph, as well as a few other similar problems.
Next is a generalisation of the Kruskal Katona Theorem [19, 17]. We are given integers k 4 we will make a lot of progress towards finding a solution.
The next chapter is to do with Turán-type problems. Given a family of k-hypergraphs F, ex(n;F) is the maximum number of edges an F-free n-vertex k-hypergraph can have. We prove that for a rational r, there exists some finite family F of k-hypergraphs for which ex(n;F) = Ɵ(nk-r) if and only if 0 < r < k - 1 or r = k.
The final chapter will deal with the implicit representation conjecture, in the special case of semi-algebraic graphs. Given a graph in such a family, we want to assign a name to each vertex in such a way that we can recover each edge based only on the names of the two incident vertices. We will first prove that one `obvious' way of storing the information doesn't work. Then we will come up with a way of storing the information that requires O(n1-E) bits per vertex, where E is some small constant depending only on the family
Two Approaches to Sidorenko's Conjecture
Sidorenko's conjecture states that for every bipartite graph on
, holds, where is the
Lebesgue measure on and is a bounded, non-negative, symmetric,
measurable function on . An equivalent discrete form of the conjecture
is that the number of homomorphisms from a bipartite graph to a graph
is asymptotically at least the expected number of homomorphisms from to the
Erd\H{o}s-R\'{e}nyi random graph with the same expected edge density as . In
this paper, we present two approaches to the conjecture. First, we introduce
the notion of tree-arrangeability, where a bipartite graph with bipartition
is tree-arrangeable if neighborhoods of vertices in have a
certain tree-like structure. We show that Sidorenko's conjecture holds for all
tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's
conjecture holds if there are two vertices in such that each
vertex satisfies or ,
and also implies a recent result of Conlon, Fox, and Sudakov \cite{CoFoSu}.
Second, if is a tree and is a bipartite graph satisfying Sidorenko's
conjecture, then it is shown that the Cartesian product of and
also satisfies Sidorenko's conjecture. This result implies that, for all , the -dimensional grid with arbitrary side lengths satisfies
Sidorenko's conjecture.Comment: 20 pages, 2 figure
Multicolor and directed edit distance
The editing of a combinatorial object is the alteration of some of its
elements such that the resulting object satisfies a certain fixed property. The
edit problem for graphs, when the edges are added or deleted, was first studied
independently by the authors and K\'ezdy [J. Graph Theory (2008), 58(2),
123--138] and by Alon and Stav [Random Structures Algorithms (2008), 33(1),
87--104]. In this paper, a generalization of graph editing is considered for
multicolorings of the complete graph as well as for directed graphs.
Specifically, the number of edge-recolorings sufficient to be performed on any
edge-colored complete graph to satisfy a given hereditary property is
investigated. The theory for computing the edit distance is extended using
random structures and so-called types or colored homomorphisms of graphs.Comment: 25 page
Homomorphisms are a good basis for counting small subgraphs
We introduce graph motif parameters, a class of graph parameters that depend
only on the frequencies of constant-size induced subgraphs. Classical works by
Lov\'asz show that many interesting quantities have this form, including, for
fixed graphs , the number of -copies (induced or not) in an input graph
, and the number of homomorphisms from to .
Using the framework of graph motif parameters, we obtain faster algorithms
for counting subgraph copies of fixed graphs in host graphs : For graphs
on edges, we show how to count subgraph copies of in time
by a surprisingly simple algorithm. This
improves upon previously known running times, such as time
for -edge matchings or time for -cycles.
Furthermore, we prove a general complexity dichotomy for evaluating graph
motif parameters: Given a class of such parameters, we consider
the problem of evaluating on input graphs , parameterized
by the number of induced subgraphs that depends upon. For every recursively
enumerable class , we prove the above problem to be either FPT or
#W[1]-hard, with an explicit dichotomy criterion. This allows us to recover
known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms
in a uniform and simplified way, together with improved lower bounds.
Finally, we extend graph motif parameters to colored subgraphs and prove a
complexity trichotomy: For vertex-colored graphs and , where is from
a fixed class , we want to count color-preserving -copies in
. We show that this problem is either polynomial-time solvable or FPT or
#W[1]-hard, and that the FPT cases indeed need FPT time under reasonable
assumptions.Comment: An extended abstract of this paper appears at STOC 201
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