Graphs whose Eulerian trails have unique labels

Abstract

Consider an undirected graph whose edges are labeled invertibly in a group. When does every Eulerian trail from one fixed vertex to another have the same label? We give a precise structural answer to this question. Essentially, we show that each ``$3$-connected part'' is labeled over a group which is isomorphic to $\mathbb{Z}_2^k$ for some $k$. We also show that the algorithmic problem admits a polynomial-time reduction to the word problem for the group.

Figures (5)

• Figure 1: The graph illustrating the Seven Bridges in Königsberg with one additional edge $e_5$ added. It has an Eulerian trail $\vec{e}_1 \vec{e}_2^{-1} \vec{e}_3 \vec{e}_4 \vec{e}_5^{-1} \vec{e}_6 \vec{e}_7 \vec{e}_8$ from $a$ to $b$. When we label the arcs $\vec{e}_i$ with elements $\alpha_i$ in a group, the label of this trail is $\alpha_1 \alpha_2^{-1} \alpha_3 \alpha_4 \alpha_5^{-1} \alpha_6 \alpha_7 \alpha_8$.
• Figure 2: Illustration of splitting off $\vec{e}_1$ and $\vec{e}_2$.
• Figure 3: A group-labeled graph $(G, \gamma)$ whose Eulerian trails from $a$ to $b$ all have the same label $(123)(12) = (13) = (12)(132)$ in the symmetric group $\mathfrak{S}_3$, but which has no shifting $\gamma'$ so that $\langle G, \gamma' \rangle\cong \mathbb{Z}_2^k$ for some $k \in \mathbb{N}$.
• Figure 4: Illustrations of $G$ and $G_e$ when $G_e \neq G-e$.
• Figure 5: An instance $(G,a,b,C)$ is illustrated on the left, where $C$ is an arbitrary Eulerian trail of $G$ from $a$ to $b$. All edges are labeled by the identity element except for the red arrowed arcs labeled by some non-identity elements in the symmetric group $\mathfrak{S}_3$. The cores of $G$ are $\{a,b\}$, $\{v_1,v_2\}$, and $\{w_1,w_2\}$. The valid instances $(H,a',b')$ obtained from these cores are illustrated on the right; we note that the precise choice of $C$ does not matter exactly because every Eulerian trail from $a$ to $b$ has the same label.

Theorems & Definitions (34)

• Theorem 1.1: Informal Statement
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Claim 2.4
• proof