Abstract
Gallai’s conjecture asserts that every connected graph on n vertices can be decomposed into at most \(\frac{n+1}{2}\) paths. The E-subgraph of a graph G, denoted by \(G_e\), is the subgraph induced by the vertices of even degree in G. A triangle pair is a graph consisting of two triangles with exactly one vertex in common. In this paper, it is proved that Gallai’s conjecture is true for graphs G in which \(G_e\) contains no triangle pair and each block of \(G_e\) has maximum degree at most 3.
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This work was supported by the National Natural Science Foundation of China (No. 11971110).
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Fan, GH., Hou, JF. & Zhou, CX. Gallai’s Conjecture on Path Decompositions. J. Oper. Res. Soc. China 11, 439–449 (2023). https://doi.org/10.1007/s40305-022-00435-3
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DOI: https://doi.org/10.1007/s40305-022-00435-3