Realizing degree sequences with $\mathcal S_3$-connected graphs

TL;DR

This work characterizes exactly which graphic sequences admit $\mathcal{S}_3$-connected realizations, tying the property to inexpensive degree conditions: $\min\{d_i\}\ge 4$ and $\sum_{i=1}^n d_i \ge 6n-4$. The authors develop a toolkit of $\mathcal{S}_3$-preserving operations (laying, lifting, and contraction) and construct both base cases and inductive reductions, using Hamiltonian closures and special graphs like $K_{(1,3,3)}$ and $K_4^*$. Consequences include that every graphic sequence with $\min\{d_i\}\ge 6$ has a realization with flow index strictly less than three, supporting Li et al.'s conjecture on $6$-edge-connected graphs. The results advance understanding of modulo-$3$ orientations and the interplay between degree sequences and strongly connected modulo-$3$ orientations.

Abstract

A graph $G$ is $\mathcal S_3$-connected if, for any mapping $β: V (G) \mapsto {\mathbb Z}_3$ with $\sum_{v\in V(G)} β(v)\equiv 0\pmod3$, there exists a strongly connected orientation $D$ satisfying $d^{+}_D(v)-d^{-}_D(v)\equiv β(v)\pmod{3}$ for any $v \in V(G)$. It is known that $\mathcal S_3$-connected graphs are contractible configurations for the property of flow index strictly less than three. In this paper, we provide a complete characterization of graphic sequences that have an $\mathcal{S}_{3}$-connected realization: A graphic sequence $π=(d_1,\, \ldots,\, d_n )$ has an $\mathcal S_3$-connected realization if and only if $\min \{d_1,\, \ldots,\, d_n\} \ge 4$ and $\sum^n_{i=1}d_i \ge 6n - 4$. Consequently, every graphic sequence $π=(d_1,\, \ldots,\, d_n )$ with $\min \{d_1,\, \ldots,\, d_n\} \ge 6$ has a realization $G$ with flow index strictly less than three. This supports a conjecture of Li, Thomassen, Wu and Zhang [European J. Combin., 70 (2018) 164-177] that every $6$-edge-connected graph has flow index strictly less than three.

Figures (6)

• Figure 1: The graphs $K_{(1,3,3)}$ and $K_4^*$.
• Figure 2: The graphs $W_4$, $(4^5,\, 3^4)$-realization, and $(6^5,\, 5^4)$-realization.
• Figure 3: The $\mathbb Z_3$-connected realizations of $(4^3,\, 3^4)$, $(4^4,\, 3^4)$, $(5,\, 4^2,\, 3^5)$, and $(5^2,\, 3^6)$.
• Figure 4: The $\mathcal{S}_3$-realizations of $(7^4,\, 4^4)$, $(7^3,\, 6,\, 5,\, 4^3)$, $(7^3,\, 5^3,\, 4^2)$, $(7^2,\, 6^3,\, 4^3)$, $(8^{3},\, 5^2,\, 4^4)$, $(8^3,\, 6,\, 4^5)$, $(9^3,\, 5,\, 4^6)$ and $(10^3,\, 4^8)$.
• Figure 5: The $\mathcal{S}_3$-connected realizations of $(6^3,\, 5^4)$, $(6^4,\, 5^4)$, $(7,\, 6^2,\, 5^5)$ and $(7^2,\, 5^6)$.

Theorems & Definitions (27)

• Conjecture 1.1: LTWZ2018
• Theorem 1.2: LXZZ2008
• Theorem 1.3: DY2016
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1: LLW2020
• Lemma 2.2: HLLW2020
• Lemma 2.3: HLLW2020
• Lemma 2.4: LLW2020
• Lemma 2.5: HLLW2020