The Hamilton–Waterloo Problem with even cycle lengths

The Hamilton–Waterloo Problem $\mathrm{HWP}(v;m,n;\alpha ,\beta )$ asks for a 2-factorization of the complete graph $K_v$ or $K_v?I$, the complete graph with the edges of a 1-factor removed, into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where $3\le m<n$. In the case that $m$ and $n$ are both even, the problem has been solved except possibly when $1\in \{\alpha ,\beta \}$ or when $\alpha$ and $\beta$ are both odd, in which case necessarily $v\equiv 2(mod4)$. In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to $\mathrm{HWP}(v;2m,2n;\alpha ,\beta )$ for odd $\alpha$ and $\beta$ whenever the obvious necessary conditions hold, except possibly if $\beta =1$; $\beta =3$ and $gcd(m,n)=1$; $\alpha =1$; or $v=2mn∕gcd(m,n)$. This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above.     

On the Hamilton–Waterloo Problem with cycle lengths of distinct parities

Let $K_v^1$ denote the complete graph $K_v$ if $v$ is odd and $K_v?I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v,M,N,\alpha ,\beta$, the Hamilton–Waterloo problem asks for a 2-factorization of $K_v^1$ into $\alpha$$C_M$-factors and $\beta$$C_N$-factors. Clearly, $M,N\ge 3$, $M\mid v$, $N\mid v$ and $\alpha +\beta =?\frac{v?1}{2}?$ are necessary conditions.Very little is known on the case where $M$ and $N$ have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever $M|N$, $v>6N>36M$, and $\beta \ge 3$.     

Orthogonally Resolvable Cycle Decompositions

If a cycle decomposition of a graph G admits two resolutions, and , such that for each resolution class and , then the resolutions and are said to be orthogonal. In this paper, we introduce the notion of an orthogonally resolvable cycle decomposition, which is a cycle decomposition admitting a pair of orthogonal resolutions. An orthogonally resolvable cycle decomposition of a graph G may be represented by a square array in which each cell is either empty or filled with a k–cycle from G, such that every vertex appears exactly once in each row and column of the array and every edge of G appears in exactly one cycle. We focus mainly on orthogonal k‐cycle decompositions of and (the complete graph with the edges of a 1‐factor removed), denoted . We give general constructions for such decompositions, which we use to construct several infinite families. We find necessary and sufficient conditions for the existence of an OCD(n, 4). In addition, we consider orthogonal cycle decompositions of the lexicographic product of a complete graph or cycle with . Finally, we give some nonexistence results.     

Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data

We study the large time behavior of viscosity solutions of Hamilton–Jacobi equations with periodic boundary data on bounded domains. We establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity.