5,618 research outputs found
Labeled Packing of Cycles and Circuits
In 2013, Duch{\^e}ne, Kheddouci, Nowakowski and Tahraoui [4, 9] introduced a
labeled version of the graph packing problem. It led to the introduction of a
new parameter for graphs, the k-labeled packing number k. This
parameter corresponds to the maximum number of labels we can assign to the
vertices of the graph, such that we will be able to create a packing of k
copies of the graph, while conserving the labels of the vertices. The authors
intensively studied the labeled packing of cycles, and, among other results,
they conjectured that for every cycle C n of order n = 2k + x, with k 2
and 1 x 2k -- 1, the value of k (C n) was 2 if x was 1
and k was even, and x + 2 otherwise. In this paper, we disprove this conjecture
by giving a counter example. We however prove that it gives a valid lower
bound, and we give sufficient conditions for the upper bound to hold. We then
give some similar results for the labeled packing of circuits
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Packing tight Hamilton cycles in 3-uniform hypergraphs
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C
\subset H is a collection of N edges for which there is an ordering of the
vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i,
v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop
new techniques which enable us to prove that under certain natural
pseudo-random conditions, almost all edges of H can be covered by edge-disjoint
tight Hamilton cycles, for N divisible by 4. Consequently, we derive the
corollary that random 3-uniform hypergraphs can be almost completely packed
with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small.
Along the way, we develop a similar result for packing Hamilton cycles in
pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur
A Pfaffian formula for monomer-dimer partition functions
We consider the monomer-dimer partition function on arbitrary finite planar
graphs and arbitrary monomer and dimer weights, with the restriction that the
only non-zero monomer weights are those on the boundary. We prove a Pfaffian
formula for the corresponding partition function. As a consequence of this
result, multipoint boundary monomer correlation functions at close packing are
shown to satisfy fermionic statistics. Our proof is based on the celebrated
Kasteleyn theorem, combined with a theorem on Pfaffians proved by one of the
authors, and a careful labeling and directing procedure of the vertices and
edges of the graph.Comment: Added referenc
Hamiltonian mappings and circle packing phase spaces: numerical investigations
In a previous paper we introduced examples of Hamiltonian mappings with phase
space structures resembling circle packings. We now concentrate on one
particular mapping and present numerical evidence which supports the conjecture
that the set of circular resonance islands is dense in phase space.Comment: 9 pages, 2 figure
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