82 research outputs found
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
Decomposing dense bipartite graphs into 4-cycles
Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles
The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)
A long-standing open question asks for the minimum number of vectors needed
to form an unextendible product basis in a given bipartite or multipartite
Hilbert space. A partial solution was found by Alon and Lovasz in 2001, but
since then only a few other cases have been solved. We solve all remaining
bipartite cases, as well as a large family of multipartite cases.Comment: 17 pages, 4 figure
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
Two Problems of Gerhard Ringel
Gerhard Ringel was an Austrian Mathematician, and is regarded as one of the most influential graph theorists of the twentieth century. This work deals with two problems that arose from Ringel\u27s research: the Hamilton-Waterloo Problem, and the problem of R-Sequences.
The Hamilton-Waterloo Problem (HWP) in the case of Cm-factors and Cn-factors asks whether Kv, where v is odd (or Kv-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. Chapter 1 gives some general constructions for such decompositions and apply them to the case where m=3 and n=3x. This problem is settle for odd v, except for a finite number of x values. When v is even, significant progress is made on the problem, although open cases are left. In particular, the difficult case of v even and s=1 is left open for many situations.
Chapter 2 generalizes the Hamilton-Waterloo Problem to complete equipartite graphs K(n:m) and shows that K(xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s,r≠1, gcd(x,z)=gcd(y,z)=1 and xyz≠0 (mod 4). Some more general constructions are given for the case when the cycles in a given two factor may have different lengths. These constructions are used to find solutions to the Hamilton-Waterloo problem for complete graphs.
Chapter 3 completes the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups
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