[Paper Review] The Hamilton-Waterloo problem for Hamilton cycles and C4k-factors

Hongchuan Lei, Hung‐Lin Fu+1 more|arXiv (Cornell University)|Sep 6, 2010

graph theory and CDMA systems3 references3 citations

TL;DR

This paper provides a complete solution to the Hamilton-Waterloo problem for the case of Hamilton cycles and $C_{4k}$-factors, proving that an $HW(n;r,s;n,4k)$ exists if and only if $r+s = ig floorrac{n-1}{2}ig floor$ and $n ot o 0 mod{4k}$ when $s > 0$, or $n o 3$ when $s = 0$. The solution relies on structured edge decompositions using cyclic group constructions and cycle factorizations over $Z_{2t} \times Z_{2k}$, with existence established for all $n \equiv 0 \pmod{4k}$ and all valid $r,s$ combinations.

ABSTRACT

In this paper we give a complete solution to the Hamilton-Waterloo problem for the case of Hamilton cycles and C4k-factors for all positive integers k.

Motivation & Objective

• To resolve the Hamilton-Waterloo problem for the case where one factor is a Hamilton cycle and the other is a $C_{4k}$-factor.
• To determine necessary and sufficient conditions for the existence of an $HW(n;r,s;n,4k)$ decomposition.
• To extend prior results on uniform 2-factorizations by handling the $C_{4k}$-factor case comprehensively.
• To provide a constructive proof using vertex partitioning and edge-matching techniques over $Z_{2t} \times Z_{2k}$.
• To establish that all feasible $r,s$ values are realizable whenever $n \equiv 0 \pmod{4k}$.

Proposed method

• Partition the vertex set of $K_n$ into $2t$ parts $V_i = \{i\} \times Z_{2k}$, where $n = 4kt$, to enable structured decomposition.
• Define edge sets $(i,j)_d = \{(i_l j_{l+d}) \mid l \in Z_{2k}\}$ as perfect matchings in the complete bipartite graph $K_{V_i,V_j}$.
• Use Lemma 2.2 to construct Hamilton cycles from sequences of matchings $(i,j)_d$ when the sum of offsets is coprime to $2k$.
• Apply Lemma 2.3 to form $C_{4k}$-factors from paired matchings when the difference of offsets is coprime to $2k$.
• Construct $F_i$ and $X$ as unions of matchings to form base subgraphs decomposable into HCs or $C_{4k}$-factors.
• Combine decompositions across $F_{2i-1} \cup F_{2i}$ and $F_1 \cup F_2 \cup F_{2t-1} \cup X$ to cover all required $r$ and $s$ values.

Experimental results

Research questions

• RQ1 For which $n$ and $k$ does an $HW(n;r,s;n,4k)$ decomposition exist?
• RQ2 Can all valid combinations of $r$ and $s$ be realized when $n \equiv 0 \pmod{4k}$?
• RQ3 What structural conditions on edge sets and offsets ensure the formation of Hamilton cycles or $C_{4k}$-factors?
• RQ4 Is the necessary condition $n \equiv 0 \pmod{4k}$ also sufficient for existence of $HW(n;r,s;n,4k)$?
• RQ5 How can the decomposition be systematically constructed for arbitrary $r$ and $s$ under the given constraints?

Key findings

• An $HW(n;r,s;n,4k)$ exists if and only if $r + s = \left\lfloor \frac{n-1}{2} \right\rfloor$ and $n \equiv 0 \pmod{4k}$ when $s > 0$, or $n \geq 3$ when $s = 0$.
• The solution is complete for all $n \equiv 0 \pmod{4k}$, with existence established for all $r \in \{0,1,\dots,\frac{n-2}{2}\}$ when $s > 0$.
• For $n = 4k$, the result follows directly from the known existence of resolvable cycle decompositions (Theorem 1.5).
• For $n = 8k$, the result is confirmed via Theorem 1.4 on $HW(n;r,s;t,2t)$, which applies when $t = 2k$.
• The set $HW^*(n;n,4k)$ contains all integers from $0$ to $\frac{n-2}{2}$ when $n \equiv 0 \pmod{4k}$, as shown by combining Propositions 3.3, 3.4, and 3.6.
• The construction method successfully realizes all $r$ and $s$ values by decomposing unions of matchings into HCs or $C_{4k}$-factors using coprimality conditions on offset sums and differences.