Partitions of multigraphs under minimum degree constraints

Schweser, Thomas; Stiebitz, Michael

doi:10.1016/j.dam.2018.10.016

Cited by 11 publications

(7 citation statements)

References 8 publications

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“…In 2017, Ban [1] proved a conclusion related to Theorem 1.1 on weighted simple graphs. Later, Schweser and Stiebitz [11] further studied this problem on graphs, and generalized the results of Stiebitz [14] and Liu and Xu [7] from simple graphs to graphs. Very recently, confirming two conjectures of Schweser and Stiebitz, Liu and Xu [8] obtained a graph version of Theorem 1.2.…”

Section: Introductionmentioning

confidence: 93%

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A generalization of Stiebitz-type results on graph decomposition

Zeng

Zu²

2020

Preprint

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In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let G be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let µis the number of edges joining u and v in G. We show that for any two functions a, b :for each x ∈ X and d Y (y) ≥ b(y) for each y ∈ Y . This extends the related results due to Diwan [3], Liu and Xu [7] and Ma and Yang [10] on simple graphs to the multigraph setting.

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Section: Introductionmentioning

confidence: 93%

“…The following proposition due to Schweser and Stiebitz [11] plays a vital role in our proof of Theorem 1.5. [11]).…”

Section: Notations and Propositionsmentioning

confidence: 98%

A generalization of Stiebitz-type results on graph decomposition

Zeng

Zu²

2020

Preprint

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“…Similarly, a partition (X, Y ) of V (G) is called an (a, b)-meager partition if X is a-meager and Y is b-meager. The following proposition due to Schweser and Stiebitz [12] plays a vital role in our proof of Theorem 5.…”

Section: Notations and Propositionsmentioning

confidence: 99%

“…Proposition 8 (Schweser and Stiebitz [12]). Let G be a graph without isolated vertices, and let a, b :…”

Section: Notations and Propositionsmentioning

confidence: 99%

A Generalization of Stiebitz-Type Results on Graph Decomposition

Zeng

2021

Electron. J. Combin.

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In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let $\mu_G(v)=\max\{\mu_G(u,v):u\in V(G)\setminus\{v\}\}$, where $\mu_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\rightarrow\mathbb{N}\setminus\{0,1\}$, if $d_G(v)\ge a(v)+b(v)+2\mu_G(v)-3$ for each $v\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such that $d_X(x)\geq a(x)$ for each $x\in X$ and $d_Y(y)\geq b(y)$ for each $y\in Y$. This extends the related results due to Diwan, Liu–Xu and Ma–Yang on simple graphs to the multigraph setting.

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“…To be precise, Diwan proved that, assuming

s, t \geq 2

, any graph of girth at least five with minimum degree at least

s + t - 1

has a partition

(A, B)

such that the subgraphs induced on A and B have minimum degree at least s and t , respectively. For related problems on graph decompositions with degree constraints or other variances, we refer readers to .…”

Section: Introductionmentioning

confidence: 99%

Decomposing C₄‐free graphs under degree constraints

Yang

2018

Journal of Graph Theory

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A celebrated theorem of Stiebitz 13 asserts that any graph with minimum degree at least s+t+1 can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This resolved a conjecture of Thomassen. In this article, we prove that for s,t≥2, if a graph G contains no cycle of length four and has minimum degree at least s+t−1, then G can be partitioned into two parts that induce two subgraphs with minimum degree at least s and t, respectively. This improves the result of Diwan in 5, where he proved the same statement for graphs of girth at least five. Our proof also works for the case of variable functions, in which the bounds are sharp as showing by some polarity graphs.

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Partitions of multigraphs under minimum degree constraints

Cited by 11 publications

References 8 publications

A generalization of Stiebitz-type results on graph decomposition

A generalization of Stiebitz-type results on graph decomposition

A Generalization of Stiebitz-Type Results on Graph Decomposition

Decomposing C₄‐free graphs under degree constraints

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Partitions of multigraphs under minimum degree constraints

Cited by 11 publications

References 8 publications

A generalization of Stiebitz-type results on graph decomposition

A generalization of Stiebitz-type results on graph decomposition

A Generalization of Stiebitz-Type Results on Graph Decomposition

Decomposing C4‐free graphs under degree constraints

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Decomposing C₄‐free graphs under degree constraints