On uniqueness of packing of three copies of 2-factors

Abstract

The packing of three copies of a graph $G$ is the union of three edge-disjoint copies (with the same vertex set) of $G$. In this paper, we completely solve the problem of the uniqueness of packing of three copies of 2-regular graphs. In particular, we show that $C_3,C_4,C_5,C_6$ and $2C_3$ have no packing of three copies, $C_7,C_8,C_3 \cup C_4, C_4 \cup C_4, C_3 \cup C_5$ and $3C_3$ have unique packing, and any other collection of cycles has at least two distinct packings.

Figures (12)

• Figure 1: The packing of three copies of $C_3 \cup C_{11}$, $C_4 \cup C_{11}$, $C_5 \cup C_{11}$ and $C_6 \cup C_{11}$ with a clique $K_5$.
• Figure 2: The packing of three copies of $C_3 \cup C_3 \cup C_{11}$ with a clique $K_5$.
• Figure 3: The extensions of packings of three copies of 2-factors from five particular families when the longest cycle has odd length.
• Figure 4: $K_5$-free packing of three copies of $C_3 \cup C_{11}$.
• Figure 5: The subgraph $H$ of the $K_5$-free packing of three copies of $C_3 \cup C_{11+4t}$ induced on $4t$ added vertices and their neighbors. Dashed edges are newly added edges, full edges are old edges.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 10
• Lemma 11
• proof