Degree Sequences vs. Forests in Finite Graphs

Abstract

We prove two conjectures of Shteiner and Shteyner stating that for an undirected graph $G=(V,E)$, the number of degree sequences arising from its spanning subgraphs is at least the number of forests in $G$, with equality if and only if $G$ is bipartite. In the process of proving the bipartite case, we provide several equivalent evaluations of the Tutte polynomial $T_G(x,y)$ at $(2,1)$, including interpretations in terms of degree vectors obtained from orientations of $G$. For the non-bipartite case, we prove strict inequality by expressing degree sequences as subset sums of signless incidence vectors and comparing these with linearly independent edge sets, showing that the presence of odd cycles yields additional independent sets beyond forests. We further strengthen this result by introducing odd pseudoforests, showing that their number is bounded above by the number of degree sequences and characterizing the corresponding independent sets accordingly.