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New Invariants for Partitioning a Graph into 2-connected Subgraphs

Michitaka Furuya, Masaki Kashima, Katsuhiro Ota

Abstract

A vertex partition in which every part induces a 2-connected subgraph is called a 2-proper partition. This concept was introduced by Ferrara et al. in 2013, and Borozan et al. gave the best possible minimum degree condition for the existence of a 2-proper partition in 2016. Later, in 2022, Chen et al. extended the result by showing a minimum degree sum condition for the existence of 2-proper partition. In this paper, we introduce two new invariants of graph, denoted by $σ^*(G)$ and $α^*(G)$. These two invariants are defined from degree sum on all independent sets with some property. We prove that if a graph $G$ satisfies $σ^*(G)\geq |V(G)|$, then with some exceptions, $G$ has a 2-proper partition with at most $α^*(G)$ parts. This result is best possible, and implies both of the results by Borozan et al. and by Chen et al.. Moreover, as a corollary of our result, we give a minimum degree product condition for the existence of a 2-proper partition.

New Invariants for Partitioning a Graph into 2-connected Subgraphs

Abstract

A vertex partition in which every part induces a 2-connected subgraph is called a 2-proper partition. This concept was introduced by Ferrara et al. in 2013, and Borozan et al. gave the best possible minimum degree condition for the existence of a 2-proper partition in 2016. Later, in 2022, Chen et al. extended the result by showing a minimum degree sum condition for the existence of 2-proper partition. In this paper, we introduce two new invariants of graph, denoted by and . These two invariants are defined from degree sum on all independent sets with some property. We prove that if a graph satisfies , then with some exceptions, has a 2-proper partition with at most parts. This result is best possible, and implies both of the results by Borozan et al. and by Chen et al.. Moreover, as a corollary of our result, we give a minimum degree product condition for the existence of a 2-proper partition.
Paper Structure (9 sections, 13 theorems, 20 equations, 4 figures)

This paper contains 9 sections, 13 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Degree, degree sum and degree product
  3. Partitioning a graph into highly connected subgraphs
  4. The minimum degree sum on large independent sets and the light independence number
  5. Main result
  6. Proof of Theorem \ref{['ind']}
  7. Fundamental properties of $\sigma^*$ and $\alpha^*$
  8. Proof of Theorem \ref{['ind']}
  9. Sharpness of Theorem \ref{['ind']}

Key Result

Theorem 1

Let $G$ be a graph of order $n$, and $k$ be an integer at least 2. If $\delta(G)\geq 2k\sqrt{n}$, then $G$ has a $k$-proper partition $\mathcal{P}$ with $|\mathcal{P}|\leq \frac{2kn}{\delta(G)}$.

Figures (4)

  • Figure 1: $F_5$
  • Figure 2: Graphs in $\mathcal{F}_{11}$
  • Figure 3: Graphs in $\mathcal{F}_{12}$
  • Figure 4: $H_{s,t}$

Theorems & Definitions (34)

  • Theorem 1: Ferrara
  • Theorem 2: Borozan
  • Conjecture 3: Borozan
  • Theorem 4: Borozan
  • Theorem 5: Chen
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 24 more