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Gallai's path decomposition conjecture states that the edges of any connected graph on n vertices can be decomposed into at most n+1 2 paths. We confirm that conjecture for all graphs with maximum degree at most five.
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the zeros of the Tutte polynomial… (More)