arXiv subjects

Nikolay Ulyanov

Publications and source records attributed to Nikolay Ulyanov.

5 recordsLinked to original sources

arXiv2026

Disproving the Petersen Coloring Conjecture: Theoretical Analysis and an Infinite Family of Counterexamples

In 1988, Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring; that is, a map $E(G) \to E(P)$ mapping any two adjacent edges of $G$ to two adjacent edges of the Petersen graph $P$. A positive resolution to Jaeger's conjecture would have immediately resolved several other famous and long-standing problems in graph theory. In July 2026, a 68-vertex counterexample was announced on X. Shortly afterwards, Putman independently presented two non-isomorphic 112-vertex counterexamples, relying solely on computer-assisted verification. In this paper, we present two counterexamples of order $52$, currently the smallest known, and provide a purely theoretical proof. In the second part, we construct an infinite family of cyclically $4$-edge-connected cubic graphs without a Petersen coloring for every even order at least $60$. Additionally, through computational verification, we show that any counterexample must have order at least $40$. Moreover, we also show that our counterexamples provide a negative answer to other related problems. Finally, we conclude the paper by discussing key open problems and highlighting avenues for future work.

math.CO↗

arXiv2026

Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture

We prove Sabidussi's compatibility conjecture. Let $G$ be a finite connected multigraph in which every vertex has even degree and the minimum degree is at least four, and let $T$ be a closed trail that traverses every edge exactly once. The edges of $G$ can be partitioned into circuits (connected 2-regular subgraphs) so that no circuit contains the two edges used consecutively anywhere in $T$. In fact, the edges can be four-coloured so that every such pair receives two different colours and the subgraph formed by the edges of each colour has even degree at every vertex. Formalization in Lean 4 is also available in the author's github.

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arXiv2026

Graph Puzzles I.1: Oriented Berge-Fulkerson Conjecture

The Berge-Fulkerson conjecture states that every bridgeless cubic graph can be covered with six perfect matchings such that each edge is covered exactly twice. An equivalent reformulation is that it's possible to find a 6-cycle 4-cover. In this paper we discuss the oriented version (o6c4c) of the latter statement, pose it as a conjecture and prove it for the family of Isaacs flower snarks. Similarly to the case of oriented cycle double cover, we can always construct an orientable surface (possibly with boundary) from an o6c4c solution. If the o6c4c solution itself splits into two (not necessarily oriented) cycle double covers, then it's also possible to build another pair of orientable surfaces (also possibly with boundaries). Finally we show how to build a ribbon graph, and for some special o6c4c cases we show that this ribbon graph corresponds to an oriented 6-cycle double cover. Github: https://github.com/gexahedron/cycle-double-covers

math.CO↗

arXiv2026

Graph Puzzles II.1: Counterexamples to Jain's Second Unit Vector Flows Conjecture

A $3$-dimensional nowhere-zero flow on a graph $G$ is a flow where each edge is assigned a $3$-dimensional vector with unit norm (which corresponds to the points of a $2$-dimensional unit sphere $S^2$). K. Jain posed two conjectures related to this idea. First one suggests that such a flow exists for all bridgeless graphs. The second conjecture states that we can assign values $\{-4,-3,-2,-1,1,2,3,4\}$ to the points of $S^2$, such that antipodal points get opposite values, and values of any three equidistant points on great circles sum to zero. If both conjectures would be true, together they would imply Tutte's 5-flow conjecture. We show 2 counterexamples to the second conjecture, by constructing sets of points each of which additionally requires values $\{-5, 5\}$. Github: https://github.com/gexahedron/unit-vector-flows

math.CO↗

arXiv2025

Open problems of the 33rd Workshop on Cycles and Colourings

Since its beginnings, every Cycles and Colourings workshop holds one or two open problem sessions; this document contains the problems (together with notes regarding the current state of the art and related bibliography) presented by participants of the 33rd edition of the workshop which took place in Nový Smokovec, Slovakia during August 31st - September 5th, 2025 (see the workshop webpage https://candc.upjs.sk).

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