Combinatorial degree version of a generalized ${\mathbb{Z}}_p$-Tucker's lemma with a combinatorial proof

Combinatorial analogues of classical Borsuk-Ulam-type theorems (e.g., Tucker’s lemma, ${\mathbb{Z}}_p$-Tucker’s lemma, etc.) have numerous important applications in combinatorics. In this paper, we formulate a combinatorial degree version of a generalized ${\mathbb{Z}}_p$-Tucker’s lemma. Our proof is purely combinatorial in the sense that it does not involve homology, cohomology or any other notions from continuous topology. In order to prove the aforementioned degree theorem, as a main technical tool, we prove a Hopf trace-type formula, which is also purely combinatorial and involves no homology. This combinatorial Hopf trace formula is of independent interest.