A tree bijection for Eulerian orientations on quartic maps

Abstract

In this paper we construct a recursive bijection between $4$-valent maps endowed with a Eulerian orientation -- also known as the ice model -- and a family of unary-binary trees. This answers a question of Bousquet-Mélou and Elvey Price, who established that both these families have the same generating function in arXiv:2503.15046 .

22 pages, 15 figures