Vertex partitioning of graphs into odd induced subgraphs

Pub Date : 2023-01-01 DOI:10.7151/dmgt.2371
A. Aashtab, S. Akbari, M. Ghanbari, Arman Shidani
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引用次数: 7

Abstract

A graph G is called an odd (even) graph if for every vertex v ∈ V (G), dG(v) is odd (even). Let G be a graph of even order. Scott in 1992 proved that the vertices of every connected graph of even order can be partitioned into some odd induced forests. We denote the minimum number of odd induced subgraphs which partition V (G) by od(G). If all of the subgraphs are forests, then we denote it by odF (G). In this paper, we show that if G is a connected subcubic graph of even order or G is a connected planar graph of even order, then odF (G) ≤ 4. Moreover, we show that for every tree T of even order odF (T ) ≤ 2 and for every unicyclic graph G of even order odF (G) ≤ 3. Also, we prove that if G is claw-free, then V (G) can be partitioned into at most ∆(G)−1 induced forests and possibly one independent set. Furthermore, we demonstrate that the vertex set of the line graph of a tree can be partitioned into at most two odd induced subgraphs and possibly one independent set.
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图的奇诱导子图的顶点划分
图G称为奇(偶)图,如果对于每个顶点v∈v (G), dG(v)是奇(偶)图。设G是偶阶的图。Scott(1992)证明了每一个偶阶连通图的顶点都可以划分成一些奇诱导林。我们表示V (G)被od(G)分割的奇诱导子图的最小数目。如果所有子图都是森林,则用odF (G)表示。本文证明了如果G是偶阶连通次三次图或G是偶阶连通平面图,则odF (G)≤4。此外,我们证明了对于每一个偶数阶odF (T)≤2的树T和每一个偶数阶odF (G)≤3的单环图G。同时,我们证明了如果G是无爪的,那么V (G)最多可以划分为∆(G)−1个诱导森林,并可能划分为一个独立集。进一步证明了树的线形图的顶点集最多可以划分为两个奇诱导子图和一个独立集。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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