1. Treewidth is NP-complete on cubic graphsHans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, Ondřej Suchý, 2025, original scientific article Abstract: The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order k also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper, we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which are related to a recent work by Chudnovsky, Norin, Seymour, and Turcotte in 2024. We also discuss several open questions.
Keywords: treewidth, cubic graph, NP'completeness
Published in RUP: 18.12.2025; Views: 145; Downloads: 0
 Full text (550,69 KB)
This document has more files! More... |
2. On cubic polycirculant nut graphsNino Bašić, Ivan Damnjanović, 2025, original scientific article Abstract: A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an $\ell$-circulant graph is a graph that admits a cyclic group of automorphisms having $\ell$ vertex orbits of equal size. It is not difficult to observe that there exists no cubic $1$-circulant nut graph or cubic $2$-circulant nut graph, while the full classification of all the cubic $3$-circulant nut graphs was recently obtained (Damnjanović et al. in Electron. J. Comb. 31(2):P2.31, 2024). Here, we investigate the existence of cubic $\ell$-circulant nut graphs for $\ell \geq 4$ and show that there is no cubic $4$-circulant nut graph or cubic $5$-circulant nut graph by using a computer-assisted proof. Furthermore, we rely on a construction based approach in order to demonstrate that there exist infinitely many cubic $\ell$-circulant nut graphs for any fixed $\ell \in \{6, 7\}$ or $\ell \geq 9$.
Keywords: nut graph, polycirculant graph, cubic graph, pregraph, voltage graph
Published in RUP: 19.11.2025; Views: 261; Downloads: 5
Full text (581,83 KB)
This document has more files! More... |
3. On regular graphs with Šoltés verticesNino Bašić, Martin Knor, Riste Škrekovski, 2025, original scientific article Abstract: Let ▫$W(G)$▫ be the Wiener index of a graph ▫$G$▫. We say that a vertex ▫$v \in V(G)$▫ is a Šoltés vertex in ▫$G$▫ if ▫$W(G - v) = W(G)$▫, i.e. the Wiener index does not change if the vertex ▫$v$▫ is removed. In 1991, Šoltés posed the problem of identifying all connected graphs ▫$G$▫ with the property that all vertices of ▫$G$▫ are Šoltés vertices. The only such graph known to this day is ▫$C_{11}$▫. As the original problem appears to be too challenging, several relaxations were studied: one may look for graphs with at least ▫$k$▫ Šoltés vertices; or one may look for ▫$\alpha$▫-Šoltés graphs, i.e. graphs where the ratio between the number of Šoltés vertices and the order of the graph is at least ▫$\alpha$▫. Note that the original problem is, in fact, to find all ▫$1$▫-Šoltés graphs. We intuitively believe that every ▫$1$▫-Šoltés graph has to be regular and has to possess a high degree of symmetry. Therefore, we are interested in regular graphs that contain one or more Šoltés vertices. In this paper, we present several partial results. For every ▫$r\ge 1$▫ we describe a construction of an infinite family of cubic ▫$2$▫-connected graphs with at least ▫$2^r$▫ Šoltés vertices. Moreover, we report that a computer search on publicly available collections of vertex-transitive graphs did not reveal any ▫$1$▫-Šoltés graph. We are only able to provide examples of large ▫$\frac{1}{3}$▫-Šoltés graphs that are obtained by truncating certain cubic vertex-transitive graphs. This leads us to believe that no ▫$1$▫-Šoltés graph other than ▫$C_{11}$▫ exists.
Keywords: Šoltés problem, Wiener index, regular graphs, cubic graphs, Cayley graph, Šoltés vertex
Published in RUP: 10.09.2025; Views: 378; Downloads: 2
Full text (456,75 KB) |
4. On balanceable and simply balanceable regular graphsMilad Ahanjideh, Martin Milanič, Mary Agnes Milanič, 2025, original scientific article Abstract: We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs G such that any 2-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of G (that is, a copy with half the edges of each color). While the problem of recognizing balanceable graphs was conjectured to be NP-complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph’s edges and another one such that the corresponding cut contains half of all the graph’s edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks’ theorem, we show that every 4-regular graph with order divisible by 4 is balanceable. Finally, we show that it is NP-complete to determine if a 9-regular graph is simply balanceable.
Keywords: balanceable graph, simply balanceable graph, cubic graph, 4-regular graph, regular graph, independent set
Published in RUP: 06.08.2025; Views: 526; Downloads: 7
Full text (530,38 KB)
This document has more files! More... |
5. |
6. Mathematical aspects of fullerenesVesna Andova, František Kardoš, Riste Škrekovski, 2016, original scientific article Abstract: Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.
Keywords: fullerene, cubic graph, planar graph, topological indices
Published in RUP: 03.01.2022; Views: 2453; Downloads: 21
Full text (626,25 KB) |
7. |
8. |
9. |
10. |
