Decomposing graphs with girth at least five under degree constraints

Abstract

We prove that the vertex set of a simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2.In [2], M. Stiebitz showed that the vertex set of a graph with minimum degree at least s + t + 1 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively. In [1], A. Kaneko showed that this result holds for triangle-free graphs with minimum degree at least s + t. In this article, we extend this result, using essentially similar ideas, and show that the bound can be further improved to s + t − 1 for graphs with girth at least 5, where s, t ≥ 2. More precisely, we prove the following.Theorem. The vertex set of any simple graph with minimum degree at least s + t − 1 and girth at least 5 can be decomposed into two parts, which induce subgraphs with minimum degree at least s and t, respectively, where s, t are positive integers ≥ 2.

A Relaxed Version of the Erdős–Lovász Tihany Conjecture

“…That this holds was conjectured in and proved by Kaneko . Divan proved that if , then for every graph containing no cycles of order 3 or 4.…”

“…That this holds was conjectured in [20] and proved by Kaneko [9]. Divan [4] proved that if s, t ≥ 2, then δ(G) ≤ s + t − 2 for every graph G ∈ G δ (s, t ) containing no cycles of order 3 or 4. Let ρ be a monotone increasing graph parameter.…”

A Note on Almost Balanced Bipartitions of a Graph

“…In [6], Stiebitz showed that every graph with minimum degree at least s + t + 1 admits an (s, t)-bipartition. Kaneko [4] and Diwan [2] strengthened this result, proving that it suffices to assume that the minimum degree is at least s + t or s + t − 1 (s, t ≥ 2) if G contains no cycles shorter than four or five, respectively.…”

“…that G admits a(2,2)-bipartition S , T such that | |S | − |T | | ≤ (n − 1)/2 − (n − 1)/2 + 5. If there exists x ∈ V(G) such that δ(G − x) ≥ 4, then δ(G ) ≥ 4 and G contains no K 3,r , as G ⊆ G, where G = G − x.…”

Decomposing C4‐free graphs under degree constraints

“…Kaneko proved that any triangle‐free graph with minimum degree at least can already force a partition as above. The minimum degree condition was further sharpen by Diwan , when cycles of length four are also forbidden. To be precise, Diwan proved that, assuming , any graph of girth at least five with minimum degree at least has a partition such that the subgraphs induced on A and B have minimum degree at least s and t , respectively.…”