On Planar Graphs Arbitrarily Decomposable into Closed Trails

Abstract

A graph G is arbitrarily decomposable into closed trails (ADCT) if the following is true: Whenever $(l_1, \ldots ,l_p)$ is a sequence of integers adding up to |E(G)| and there is a closed trail of length l _i in G for i = 1, ... , p, then there is a sequence (T₁, ... , T _p ) of pairwise edge-disjoint closed trails in G such that T _i is of length l _i for $i=1, \ldots ,p$. In the paper it is proved that a 2n-vertex bipyramid is ADCT for any integer n ≥ 3. Further, if G is a 4-connected planar graph that is ADCT, it contains at most four edges incident only with faces of degree at least 4. There are examples showing that the bound of four edges is tight.