We prove that the recognition of exact cubes of trees can be done in polynomial time. More precisely, the exact distance power of a graph is a refinement of the more usual notion of graph power. Given a graph G and a positive integer p , the exact distance p th power of G is the graph G #p on the same vertex set where two vertices are adjacent if their distance is exactly p in G . Recently Bai et al. [Y. Bai, P. P. Cortés, R. Naserasr and D. A. Quiroz. Characterizing and recognizing exact-distance squares of graphs. Discrete Mathematics 347(8). 2024] proved that the recognition of exact squares of trees is polynomially tractable. In order to extend this result to exact cubes of trees, we first test whether there is a caterpillar as an exact cubic root and, if not, proceed with a general tree. Both algorithms rely on the observation that the knowledge of a fixed number of vertices is roughly enough to deduce the whole structure of the tree we aim for.

The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains K X(G) as an immersion. Our work is motivated by a strengthening of this conjecture which asserts that every graph G contains K X(G) as a totally odd immersion. As evidence for this strengthened conjecture and inspired by a result of Steiner (2024), we show that if the chromatic number of G is equal to any of its topological lower bounds, then G contains a totally odd immersion of K [x( G)/2] +1 . Kneser graphs are canonical examples of graphs satisfying such equalities for their chromatic numbers. Simonyi and Zsban (2010) showed that every Kneser graph G with large enough order (compared to x (G) ) contains a totally odd subdivision of K X(G) , thus satisfying the motivating conjecture in a strong sense. We show that, in fact, for every t ≥ 8, there are t -chromatic Kneser graphs that contain arbitrarily large complete totally odd subdivisions.

The counterexamples that Catlin used to disprove Hajós’ conjecture (For every integer t ≥ 0, every graph G with no subdivision of K t+1 is t- colorable.) are the lexicographic product of cliques and cycles. In 1989, Lescure and Meyniel made a conjecture that is a weakening of Hajós’ and that remains open: For every integer t ≥ 0, every graph G with no immersion of K t+1 is t- colorable. Can minimal counterexamples to this conjecture be produced through graph products? Collins, Heenehan, and McDonald recently gave a negative answer to this question for the lexicographic and Cartesian product. We consider a strengthening of the Lescure and Meyniel conjecture, different from that of Hajós, based on the notion of totally odd immersions. We study the largest totally odd immersion appearing in the graph product of two graphs. Our results imply that no minimal counterexample to this strengthened conjecture can be obtained from the Cartesian, lexicographic, direct (tensor) or strong product of graphs.