104 research outputs found
Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings
We show a surprising link between experimental setups to realize
high-dimensional multipartite quantum states and Graph Theory. In these setups,
the paths of photons are identified such that the photon-source information is
never created. We find that each of these setups corresponds to an undirected
graph, and every undirected graph corresponds to an experimental setup. Every
term in the emerging quantum superposition corresponds to a perfect matching in
the graph. Calculating the final quantum state is in the complexity class
#P-complete, thus cannot be done efficiently. To strengthen the link further,
theorems from Graph Theory -- such as Hall's marriage problem -- are rephrased
in the language of pair creation in quantum experiments. We show explicitly how
this link allows to answer questions about quantum experiments (such as which
classes of entangled states can be created) with graph theoretical methods, and
potentially simulate properties of Graphs and Networks with quantum experiments
(such as critical exponents and phase transitions).Comment: 6+5 pages, 4+7 figure
A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs
The Hamilton-Waterloo problem asks for which and the complete graph
can be decomposed into copies of a given 2-factor and
copies of a given 2-factor (and one copy of a 1-factor if is even).
In this paper we generalize the problem to complete equipartite graphs
and show that can be decomposed into copies of a
2-factor consisting of cycles of length ; and copies of a 2-factor
consisting of cycles of length , whenever is odd, ,
and . We also give some more general
constructions where the cycles in a given two factor may have different
lengths. We use these constructions to find solutions to the Hamilton-Waterloo
problem for complete graphs
Cycle-factorization of symmetric complete multipartite digraphs
AbstractFirst, we show that a necessary and sufficient condition for the existence of a C3-factorization of the symmetric tripartite digraph Kn1,n2,n3∗, is n1 = n2 = n3. Next, we show that a necessary and sufficient condition for the existence of a C̄2k-factorization of the symmetric complete multipartite digraph Kn1, n2,…,nm is n1 = n2 = … = nm = 0 (mod k) for even m and n1 = n2 = … = ≡ 0 (mod 2k) for odd m
Star-factorization of symmetric complete bipartite multi-digraphs
AbstractWe show that a necessary and sufficient condition for the existence of an Sk-factorization of the symmetric complete bipartite multi-digraph λKm,n∗ is m=n≡0(modk(k−1)/d), where d=(λ,k−1)
Multipartite graph decomposition: cycles and closed trails
This paper surveys results on cycle decompositions of complete multipartite graphs (where the parts are not all of size 1, so the graph is not K_n ), in the case that the cycle lengths are “small”. Cycles up to length n are considered, when the complete multipartite graph has n parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned
Cyclic cycle systems of the complete multipartite graph
In this paper, we study the existence problem for cyclic -cycle
decompositions of the graph , the complete multipartite graph with
parts of size , and give necessary and sufficient conditions for their
existence in the case that
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
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