Multigraphs with Unique Partition into Cycles

TL;DR

The paper addresses when Eulerian multigraphs and digraphs admit a unique edge-disjoint partition into cycles, introducing the closure class $\mathcal{S}$ under the operation $D_1 \ast D_2$. It proves that a connected Eulerian digraph has a unique partition into cycles if and only if it lies in $\mathcal{S}$ (i.e., is a bridgeless cactus digraph), and it further provides a suite of equivalent conditions for having a unique Eulerian circuit, including a max out-degree bound $\Delta^{+}(D)\le 2$ and a tree-like block intersection $G_{\mathcal{B}(D)}$. The results identify Christmas cactus digraphs as the precise subfamily with a unique Eulerian circuit and extend the characterization to loop-allowed digraphs. The undirected analogue is developed, showing that a connected Eulerian multigraph $X$ has a unique partition into undirected cycles exactly when $X\in\mathcal{S}$, with the cyclomatic number $\mathbb{C}(X)$ equaling the number of cycles $|\mathcal{F}(X)|$ and $\mathcal{F}(X)$ forming a partition of $E(X)$.

Abstract

Due to Veblen's Theorem, if a connected multigraph $X$ has even degrees at each vertex, then it is Eulerian and its edge set has a partition into cycles. In this paper, we show that an Eulerian multigraph has a unique partition into cycles if and only if it belongs to the family $\mathcal{S}$, ``bridgeless cactus multigraphs", elements of which are obtained by replacing every edge of a tree with a cycle of length $\geq 2$. Other characterizing conditions for bridgeless cactus multigraphs and digraphs are provided. Furthermore, for a digraph $D$, we list conditions equivalent to having a unique Eulerian circuit, thereby generalizing a previous result of Arratia-Bollobás-Sorkin. In particular, we show that digraphs with a unique Eulerian circuit constitute a subfamily of $\mathcal{S}$, namely, ``Christmas cactus digraphs".

Figures (4)

• Figure 1: An element of the collection $\mathcal{S}$.
• Figure 2: Two cycles $\{\beta_i\}_{i=1,2}$ intersecting at $u\neq v$ and the partitioning digraphs $\{D_i\}_{i=1,2}$
• Figure 3: Pairwise intersecting cycles $\{\beta_1,\beta_2,\beta_t\}$ and the alternative partition
• Figure 4: Alternative partition of $D_i \cup \beta_t$ in a special case of the proof of \ref{['lem:unique_decomp_equiv_treelike']}

Theorems & Definitions (35)

• Definition 1
• Definition 2
• Definition 3
• Theorem 2.1: B.E.S.T. Theorem
• Definition 4
• Remark 1
• Remark 2
• Definition 5
• Definition 6
• Remark 3