A Generalization of the Hamilton-Waterloo Problem on Complete Equipartite Graphs

The Hamilton-Waterloo problem asks for which $s$ and $r$ the complete graph $K_n$ can be decomposed into $s$ copies of a given 2-factor $F_1$ and $r$ copies of a given 2-factor $F_2$ (and one copy of a 1-factor if $n$ is even). In this paper we generalize the problem to complete equipartite graphs $K_{(n:m)}$ and show 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, $s,r\neq 1$, $\gcd(x,z)=\gcd(y,z)=1$ and $xyz\neq 0 \pmod 4$. 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

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

Clique decompositions of multipartite graphs and completion of Latin squares

Our main result essentially reduces the problem of finding an edge-decomposition of a balanced r-partite graph of large minimum degree into r-cliques to the problem of finding a fractional r-clique decomposition or an approximate one. Together with very recent results of Bowditch and Dukes as well as Montgomery on fractional decompositions into triangles and cliques respectively, this gives the best known bounds on the minimum degree which ensures an edge-decomposition of an r-partite graph into r-cliques (subject to trivially necessary divisibility conditions). The case of triangles translates into the setting of partially completed Latin squares and more generally the case of r-cliques translates into the setting of partially completed mutually orthogonal Latin squares.Comment: 40 pages. To appear in Journal of Combinatorial Theory, Series

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 <em>K</em>_<em>n</em> ), in the case that the cycle lengths are “small”. Cycles up to length <em>n</em> are considered, when the complete multipartite graph has <em>n</em> parts, but not hamilton cycles. Properties which the decompositions may have, such as being gregarious, are also mentioned.<br /

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

Some Implications on Amorphic Association Schemes

AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition