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Keywords:

  • graph;
  • girth;
  • path-decomposition

Abstract

In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path-decomposition of G having at most inline image paths. In this article, we show that for any simple graph G having girth inline image, there is a path-decomposition of G having at most inline image paths, where inline image is the number of vertices of odd degree in G and inline image is the number of nonisolated vertices of even degree in G.