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Decompositions into two linear forests of bounded lengths

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Title
Decompositions into two linear forests of bounded lengths
Author(s)
Rutger Campbell; Hörsch, Florian; Moore, Benjamin
Publication Date
2024-06
Journal
Discrete Mathematics, v.347, no.6
Publisher
Elsevier BV
Abstract
For some k∈Z≥0∪{∞}, we call a linear forest k-bounded if each of its components has at most k edges. We will say a (k,ℓ)-bounded linear forest decomposition of a graph G is a partition of E(G) into the edge sets of two linear forests Fk,Fℓ where Fk is k-bounded and Fℓ is ℓ-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both k and ℓ are at least 2, NP-complete if k≥9 and ℓ=1, and is in P for (k,ℓ)=(2,1). Before this, the only known NP-complete cases were the (2,2) and (3,3) cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than 3-edge-coloring such graphs. © 2024 Elsevier B.V.
URI
https://pr.ibs.re.kr/handle/8788114/15028
DOI
10.1016/j.disc.2024.113962
ISSN
0012-365X
Appears in Collections:
Pioneer Research Center for Mathematical and Computational Sciences(수리 및 계산과학 연구단) > Discrete Mathematics Group(이산 수학 그룹) > 1. Journal Papers (저널논문)
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