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Gallai’s Path Decomposition Conjecture for Graphs with Small Bowtie Boundaries

Idris Ali Shaik

Palomar Registrypalomar/PALOMAR-2026-09-14-000005Sep 14, 2026math.COscore 0

We prove sufficient conditions for partitioning the edges of a connected finite simple graph into at most ⌈|V(G)|/2⌉ simple paths. The results concern bowtie components—two triangles sharing exactly one vertex—in the subgraph induced by even-degree vertices. They include removal preserving specified endpoints, a sharp full-incidence star expansion, composition under explicit boundary hypotheses, and corollaries for graphs with two bowtie components. The unrestricted two-exception conjecture remains open.

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