arXiv subjects

Radek Hušek

Publications and source records attributed to Radek Hušek.

5 recordsLinked to original sources

arXiv2026

Exponentially Many Circuit Double Covers

The cycle double cover conjecture of Szekeres and Seymour, the proof of which was recently announced by OpenAI, states that every bridgeless graph has a collection of cycles covering every edge exactly twice. We study the counting version of this statement for cubic graphs, where we count circuit double covers --- collections of circuits (connected 2-regular subgraphs) covering every edge twice. We show that every 2-edge-connected 3-edge-colorable cubic graph on $n$ vertices has at least $2^{n/2-1}$ circuit double covers, matching our previously conjectured general lower bound. For every 3-edge-connected cubic graph with girth at least 16 we show a weaker exponential lower bound on circuit double covers. For both of these results we use the same system of linear equations used by OpenAI in their proof, however, we provide additional combinatorial interpretation. We characterize planarity of a cubic graph by solvability of this system of equations for arbitrary nowhere-zero $\mathbb Z_2^k$-flow. We give a condition on the flow that is equivalent to existence of a 5-cycle double cover.

math.CO↗

arXiv2024

Counting Circuit Double Covers

We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an almost-exponential lower-bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower-bound for planar graphs. We conjecture that any bridgeless cubic graph has at least $2^{n/2-1}$ circuit double covers and we show an infinite class of graphs for which this bound is tight.

math.CO↗

arXiv2020

Approximation Algorithms for Steiner Tree Based on Star Contractions: A Unified View

In the Steiner Tree problem we are given an edge weighted undirected graph $G = (V,E)$ and a set of terminals $R \subseteq V$. The task is to find a connected subgraph of $G$ containing $R$ and minimizing the sum of weights of its edges. We observe that many approximation algorithms for Steiner Tree follow a similar scheme (meta-algorithm) and perform (exhaustively) a similar routine which we call star contraction. Here, by a star contraction, we mean finding a star-like subgraph in the input graph minimizing the ratio of its weight to the number of contained terminals minus one. It is not hard to see that the well-known MST-approximation seeks the best star to contract among those containing two terminals only. We perform an empirical study of star contractions with the relaxed condition on the number of terminals in each star contraction. Our experiments suggest the following: -- if the algorithm performs star contractions exhaustively, the quality of the solution is usually slightly better than Zelikovsky's 11/6-approximation algorithm, -- on average the quality of the solution returned by the MST-approximation algorithm improves with every star contraction, -- the same holds for iterated MST (MST+), which outperforms MST in every measurement while keeping very fast running time (on average $\sim 3\times$ slower than MST), -- on average the quality of the solution obtained by exhaustively performing star contraction is about 16\% better than the initial MST-approximation, and -- we propose a more precise way to find the so-called improved stars which yield a slightly better solution within a comparable running time (on average $\sim 3\times$ slower). Furthermore, we propose two improvements of Zelikovsky's 11/6-approximation algorithm and we empirically confirm that the quality of the solution returned by any of these is better than the one returned by the former algorithm.

cs.DS↗

arXiv2019

Homomorphisms of Cayley graphs and Cycle Double Covers

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from Cayley graph Cay(M, B) to another Cayley graph Cay(M', B') then every graph with an (M, B)-flow has an (M', B')-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in all cases but we conjecture that it still holds for an interesting subclass of them and we prove a partial result in this direction. We also show that the original conjecture implies the existence of an oriented cycle double cover with a small number of cycles.

math.CO↗

arXiv2017

Group Connectivity: $\mathbb Z_4$ v. $\mathbb Z_2^2$

We answer a question on group connectivity suggested by Jaeger et al. [Group connectivity of graphs -- A nonhomogeneous analogue of nowhere-zero flow properties, JCTB 1992]: we find that $\mathbb Z_2^2$-connectivity does not imply $\mathbb Z_4$-connectivity, neither vice versa. We use a computer to find the graphs certifying this and to verify their properties using non-trivial enumerative algorithm. While the graphs are small (the largest has 15 vertices and 21 edges), a computer-free approach remains elusive.

cs.DM↗