Minimum degree conditions for monochromatic cycle partitioning

A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.Comment: 22 pages (26 including appendix

A study of 4−4-cycle systems

A $4-$cycle system is a partition of the edges of the complete graph $K_n$ into $4-$cycles. Let ${ C}$ be a collection of cycles of length 4 whose edges partition the edges of $K_n$. A set of 4-cycles $T_1 \subset C$ is called a 4-cycle trade if there exists a set $T_2$ of edge-disjoint 4-cycles on the same vertices, such that $({C} \setminus T_1)\cup T_2$ also is a collection of cycles of length 4 whose edges partition the edges of $K_n$. We study $4-$cycle trades of volume two (double-diamonds) and three and show that the set of all 4-CS(9) is connected with respect of trading with trades of volume 2 (double-diamond) and 3. In addition, we present a full rank matrix whose null-space is containing trade-vectors

Hajós' conjecture and small cycle double covers of planar graphs

AbstractWe prove that every simple even planar graph on n vertices has a partition of its edge set into at most ⌊(n - 1)/2⌋ cycles. A previous proof of this result was given by Tao, but is incomplete, and we provide here a somewhat different proof. We also discuss the connection between this result and the Small Cycle Double Cover Conjecture

The number of Hamiltonian decompositions of regular graphs

A Hamilton cycle in a graph $\Gamma$ is a cycle passing through every vertex of $\Gamma$. A Hamiltonian decomposition of $\Gamma$ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki's theorem from the 19th century, showing that a complete graph $K_n$ on an odd number of vertices $n$ has a Hamiltonian decomposition. This result was recently greatly extended by K\"{u}hn and Osthus. They proved that every $r$-regular $n$-vertex graph $\Gamma$ with even degree $r=cn$ for some fixed $c>1/2$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently large. In this paper we address the natural question of estimating $H(\Gamma)$, the number of such decompositions of $\Gamma$. Our main result is that $H(\Gamma)=r^{(1+o(1))nr/2}$. In particular, the number of Hamiltonian decompositions of $K_n$ is $n^{(1-o(1))n^2/2}$

Cycle partitions of regular graphs

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$, improving a result of Han, who showed that in this range almost all vertices of $G$ can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact, our proof gives a partition of $V(G)$ into cycles. We also show that, if $d = \Omega(n)$ and $G$ is bipartite, then $V(G)$ can be partitioned into $n / (2d)$ paths (this bound in tight for bipartite graphs).Comment: 31 pages, 1 figur

Single-Strip Triangulation of Manifolds with Arbitrary Topology

Triangle strips have been widely used for efficient rendering. It is NP-complete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50% in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a single-strip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200