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.
Given a minor-closed class of graphs G, what is the infimum of the non-trivial roots of the chromatic polynomial of G ∈ G? When G is the class of all graphs, the answer is known to be 32/27. We answer this question exactly for three minorclosed classes of graphs. Furthermore, we conjecture precisely when the value is larger than 32/27.