Affiliation:
1. School of Mathematics and Statistics, Ningbo University, Ningbo, China
Abstract
Gallai conjectured that every connected graph on n vertices can be decomposed
into at most n+1/2 paths. Let G be a connected graph on n vertices. The
E-subgraph of G, denoted by F, is the subgraph induced by the vertices of
even degree in G. The maximum degree of G is denoted by ?(G). In 2020,
Botler and Sambinelli verified Gallai?s Conjecture for graphs whose
E-subgraphs F satisfy ?(F) ? 3. If the E-subgraph of G has at most one
vertex with degree greater than 3, Fan, Hou and Zhou verified Gallai?s
Conjecture for G. In this paper, it is proved that if there are two adjacent
vertices x, y ? V(F) such that dF(v) ? 3 for every vertex v ? V(F)\{x, y},
then G has a path-decomposition D1 such that |D1| ? n+1/2 and D1(x) ? 2, and
a path-decomposition D2 such that |D2| ? n+1/2 and D2(y) ? 2.
Publisher
National Library of Serbia
Reference7 articles.
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4. G. Fan, J. Hou and C. Zhou, Gallai’s conjecture on path decompositions, Journal of the operations research society of China (2022), accepted.
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