arXiv subjects

Jan Goedgebeur

Publications and source records attributed to Jan Goedgebeur.

At least 37 records · Page 2Linked to original sources

arXiv2024

The Frank number and nowhere-zero flows on graphs

An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. Inspired by a problem of Frank, in 2021 Hörsch and Szigeti proposed a new parameter for $3$-edge-connected graphs, called the Frank number, which refines $k$-edge-connectivity. The Frank number is defined as the minimum number of orientations of $G$ for which every edge of $G$ is deletable in at least one of them. They showed that every $3$-edge-connected graph has Frank number at most $7$ and that in case these graphs are also $3$-edge-colourable the parameter is at most $3$. Here we strengthen both results by showing that every $3$-edge-connected graph has Frank number at most $4$ and that every graph which is $3$-edge-connected and $3$-edge-colourable has Frank number $2$. The latter also confirms a conjecture by Barát and Blázsik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number $2$ and use them in an algorithm to computationally show that the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices having Frank number greater than $2$.

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arXiv2024

Infinite families of planar graphs of a given injective chromatic number

An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud and Wang conjectured that every planar graph of maximum degree $Δ\ge 3$ admits an injective colouring with at most $\lfloor 3Δ/2\rfloor$ colours. This was later disproved by Lužar and Škrekovski for certain small and even values of $Δ$ and they proposed a new refined conjecture. Using an algorithm for determining the injective chromatic number of a graph, i.e. the smallest number of colours for which the graph admits an injective colouring, we give computational evidence for Lužar and Škrekovski's conjecture and extend their results by presenting an infinite family of $3$-connected planar graphs for each $Δ$ (except for $4$) attaining their bound, whereas they only gave a finite amount of examples for each $Δ$. Hence, together with another infinite family of maximum degree $4$, we provide infinitely many counterexamples to the conjecture by Chen et al. for each $Δ$ if $4\le Δ\le 7$ and every even $Δ\ge 8$. We provide similar evidence for analogous conjectures by La and Štorgel and Lužar, Škrekovski and Tancer when the girth is restricted as well. Also in these cases we provide infinite families of $3$-connected planar graphs attaining the bounds of these conjectures for certain maximum degrees $Δ\geq 3$.

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arXiv2024

Computational methods for finding bi-regular cages

An $(\{r,m\};g)$-graph is a (simple, undirected) graph of girth $g\geq3$ with vertices of degrees $r$ and $m$ where $2 \leq r < m$ . Given $r,m,g$, we seek the $(\{r,m\};g)$-graphs of minimum order, called $(\{r,m\};g)$-cages or bi-regular cages, whose order is denoted by $n(\{r,m\};g)$. In this paper, we use computational methods for finding $(\{r,m\};g)$-graphs of small order. Firstly, we present an exhaustive generation algorithm, which leads to $\unicode{x2013}$ previously unknown $\unicode{x2013}$ exhaustive lists of $(\{r,m\};g)$-cages for 24 different triples $(r,m,g)$. This also leads to the improvement of the lower bound of $n(\{4,5\};7)$ from 66 to 69. Secondly, we improve 49 upper bounds of $n(\{r,m\};g)$ based on constructions that start from $r$-regular graphs. Lastly, we generalize a theorem by Aguilar, Araujo-Pardo and Berman [arXiv:2305.03290, 2023], leading to 73 additional improved upper bounds.

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arXiv2024

Cubic graphs with edges in exactly one perfect matching

Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. Actually, it is known that for a very large fraction of bridgeless cubic graphs, every edge belongs to at least two distinct perfect matchings. In this paper, we study the class of non-double covered cubic graphs, i.e.\ graphs having an edge, called lonely edge, which belongs to exactly one perfect matching. First of all, we provide a reduction of the problem to the subclass $\cal U$ of $3$-connected cubic graphs. Then, we furnish an inductive characterization of $\cal U$ and we study properties related to the count of lonely edges. In particular, denoting by $\mathcal{U}_k$ the subclass of graphs of $\cal U$ with exactly $k$ lonely edges, we prove that $\mathcal{U}_k$ is empty for $k>6$, and we present a complete characterization for $3 \leq k \leq 6$. The paper concludes with some insights on ${\cal U}_1$ and ${\cal U}_2$.

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arXiv2024

The maximum number of connected sets in regular graphs

We improve the best known lower bounds on the exponential behavior of the maximum of the number of connected sets, $N(G)$, and dominating connected sets, $N_{dom}(G)$, for regular graphs. These lower bounds are improved by constructing a family of graphs defined in terms of a small base graph (a Moore graph), using a combinatorial reduction of these graphs to rectangular boards followed by using linear algebra to show that the lower bound is related to the largest eigenvalue of a coefficient matrix associated with the base graph. We also determine the exact maxima of $N(G)$ and $N_{dom}(G)$ for cubic and quartic graphs of small order. We give multiple results in favor of a conjecture that each Moore graph $M$ maximizes the base indicating the exponential behavior of the number of connected vertex subsets among graphs with at least $|M|$ vertices and the same regularity. We improve the best known upper bounds for $N(G)$ and $N_{dom}(G)$ conditional on this conjecture.

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arXiv2024

On graphs isomorphic with their conduction graph

Conduction graphs are defined here in order to elucidate at a glance the often complicated conduction behaviour of molecular graphs as ballistic molecular conductors. The graph $G^{\mathrm C}$ describes all possible conducting devices associated with a given base graph $G$ within the context of the Source-and-Sink-Potential model of ballistic conduction. The graphs $G^{\mathrm C}$ and $G$ have the same vertex set, and each edge $xy$ in $G^{\mathrm C}$ represents a conducting device with graph $G$ and connections $x$ and $y$ that conducts at the Fermi level. If $G^{\mathrm C}$ is isomorphic with the simple graph $G$ (in which case we call $G$ conduction-isomorphic), then $G$ has nullity $η(G)=0$ and is an ipso omni-insulator. Motivated by this, examples are provided of ipso omni-insulators of odd order, thereby answering a recent question. For $η(G)=0$, $G^{\mathrm C}$ is obtained by 'booleanising' the inverse adjacency matrix $A^{-1}(G)$, to form $A(G^{\mathrm C})$, i.e. by replacing all non-zero entries $(A(G)^{-1})_{xy}$ in the inverse by $1+δ_{xy}$ where $δ_{xy}$ is the Kronecker delta function. Constructions of conduction-isomorphic graphs are given for the cases of $G$ with minimum degree equal to two or any odd integer. Moreover, it is shown that given any connected non-bipartite conduction-isomorphic graph $G$, a larger conduction-isomorphic graph $G'$ with twice as many vertices and edges can be constructed. It is also shown that there are no 3-regular conduction-isomorphic graphs. A census of small (order $\leq 11$) connected conduction-isomorphic graphs and small (order $\leq 22$) connected conduction-isomorphic graphs with maximum degree at most three is given. For $η(G)=1$, it is shown that $G^{\mathrm C}$ is connected if and only if $G$ is a nut graph (a singular graph of nullity one that has a full kernel vector).

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arXiv2024

$K_2$-Hamiltonian Graphs: II

In this paper we use theoretical and computational tools to continue our investigation of $K_2$-hamiltonian graphs, that is, graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, and their interplay with $K_1$-hamiltonian graphs, that is, graphs in which every vertex-deleted subgraph is hamiltonian. Perhaps surprisingly, there exist graphs that are both $K_1$- and $K_2$-hamiltonian, yet non-hamiltonian, for example, the Petersen graph. Grünbaum conjectured that every planar $K_1$-hamiltonian graph must itself be hamiltonian; Thomassen disproved this conjecture. Here we show that even planar graphs that are both $K_1$- and $K_2$-hamiltonian need not be hamiltonian, and that the number of such graphs grows at least exponentially. Motivated by results of Aldred, McKay, and Wormald, we determine for every integer $n$ that is not 14 or 17 whether there exists a $K_2$-hypohamiltonian, that is, non-hamiltonian and $K_2$-hamiltonian, graph of order $n$, and characterise all orders for which such cubic graphs and such snarks exist. We also describe the smallest cubic planar graph which is $K_2$-hypohamiltonian, as well as the smallest planar $K_2$-hypohamiltonian graph of girth $5$. We conclude with open problems and by correcting two inaccuracies from the first article.

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arXiv2024

Exhaustive generation of edge-girth-regular graphs

Edge-girth-regular graphs (abbreviated as $egr$ graphs) are a class of highly regular graphs. More specifically, for integers $v$, $k$, $g$ and $λ$ an $egr(v,k,g,λ)$ graph is a $k$-regular graph with girth $g$ on $v$ vertices such that every edge is contained in exactly $λ$ cycles of length $g$. The central problem in this paper is determining $n(k,g,λ)$, which is defined as the smallest integer $v$ such that an $egr(v,k,g,λ)$ graph exists (or $\infty$ if no such graph exists) as well as determining the corresponding extremal graphs. We propose a linear time algorithm for computing how often an edge is contained in a cycle of length $g$, given a graph with girth $g$. We use this as one of the building blocks to propose another algorithm that can exhaustively generate all $egr(v,k,g,λ)$ graphs for fixed parameters $v, k, g$ and $λ$. We implement this algorithm and use it in a large-scale computation to obtain several new extremal graphs and improvements for lower and upper bounds from the literature for $n(k,g,λ)$. Among others, we show that $n(3,6,2)=24, n(3,8,8)=40, n(3,9,6)=60, n(3,9,8)=60, n(4,5,1)=30, n(4,6,9)=35, n(6,5,20)=42$ and we disprove a conjecture made by Araujo-Pardo and Leemans [Discrete Math. 345(10):112991 (2022)] for the cubic girth 8 and girth 12 cases. Based on our computations, we conjecture that $n(3,7,6)=n(3,8,10)=n(3,8,12)=n(3,8,14)=\infty.$

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arXiv2024

Exact enumeration of fullerenes

A fullerene, or buckyball, is a trivalent graph on the sphere with only pentagonal and hexagonal faces. Building on ideas of Thurston, we use modular forms to give an exact formula for the number of oriented fullerenes with a given number of vertices.

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arXiv2024

Some Results on Critical ($P_5,H$)-free Graphs

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$, but $G$ has chromatic number $k$. The study of $k$-vertex-critical graphs for specific graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there exists a polynomial-time certifying algorithm to decide the $k$-colorability of a graph in the class. In this paper, we show that: (1) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,K_{1,4}+P_1)$-free graphs; (2) for $s \ge 1$, there are finitely many 5-vertex-critical $(P_5,K_{1,s}+P_1)$-free graphs; (3) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,\overline{K_3+2P_1})$-free graphs. Moreover, we characterize all $5$-vertex-critical $(P_5,H)$-free graphs where $H \in \{K_{1,3}+P_1,K_{1,4}+P_1,\overline{K_3+2P_1}\}$ using an exhaustive graph generation algorithm.

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arXiv2024

Generation and New Infinite Families of $K_2$-hypohamiltonian Graphs

We present an algorithm which can generate all pairwise non-isomorphic $K_2$-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for $K_2$-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which $K_2$-hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite $K_2$-hypohamiltonian graphs. Furthermore, we describe a new operation for creating $K_2$-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar $K_2$-hypohamiltonian graph of order $n$ for every integer $n\geq 134$. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence $K_2$-hypohamiltonian graphs with large maximum degree and size.

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arXiv2023

A heuristic algorithm using tree decompositions for the maximum happy vertices problem

We propose a new methodology to develop heuristic algorithms using tree decompositions. Traditionally, such algorithms construct an optimal solution of the given problem instance through a dynamic programming approach. We modify this procedure by introducing a parameter $W$ that dictates the number of dynamic programming states to consider. We drop the exactness guarantee in favour of a shorter running time. However, if $W$ is large enough such that all valid states are considered, our heuristic algorithm proves optimality of the constructed solution. In particular, we implement a heuristic algorithm for the Maximum Happy Vertices problem using this approach. Our algorithm more efficiently constructs optimal solutions compared to the exact algorithm for graphs of bounded treewidth. Furthermore, our algorithm constructs higher quality solutions than state-of-the-art heuristic algorithms Greedy-MHV and Growth-MHV for instances of which at least 40\% of the vertices are initially coloured, at the cost of a larger running time.

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arXiv2023

Critical $(P_5,dart)$-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. Let $P_t$ be the path on $t$ vertices. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond. In this paper, we show that there are finitely many $k$-vertex-critical $(P_5,dart)$-free graphs for $k \ge 1$ To prove these results, we use induction on $k$ and perform a careful structural analysis via Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for $k \in \{5, 6, 7\}$ we characterize all $k$-vertex-critical $(P_5,dart)$-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k \ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.

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arXiv2022

Few hamiltonian cycles in graphs with one or two vertex degrees

We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe's computational enumerative results from [Experim. Math. 27 (2018) 426-430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen's independent dominating set method. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen. Motivated by an observation of Aldred and Thomassen, we prove that for every $κ\in \{ 2, 3 \}$ and any positive integer $k$, there are infinitely many non-regular graphs of connectivity $κ$ containing exactly one hamiltonian cycle and in which every vertex has degree $3$ or $2k$.

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arXiv2022

House of Graphs 2.0: a database of interesting graphs and more

In 2012 we announced the House of Graphs (https://houseofgraphs.org) [Discrete Appl. Math. 161 (2013), 311-314], which was a new database of graphs. The House of Graphs hosts complete lists of graphs of various graph classes, but its main feature is a searchable database of so called "interesting" graphs, which includes graphs that already occurred as extremal graphs or as counterexamples to conjectures. An important aspect of this database is that it can be extended by users of the website. Over the years, several new features and graph invariants were added to the House of Graphs and users uploaded many interesting graphs to the website. But as the development of the original House of Graphs website started in 2010, the underlying frameworks and technologies of the website became outdated. This is why we completely rebuilt the House of Graphs using modern frameworks to build a maintainable and expandable web application that is future-proof. On top of this, several new functionalities were added to improve the application and the user experience. This article describes the changes and new features of the new House of Graphs website.

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arXiv2021

The Minimality of the Georges-Kelmans Graph

In 1971, Tutte wrote in an article that "it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian". Motivated by this remark, Horton constructed a counterexample on 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts -- even if they have girth 6 -- it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs. In 1969, Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette's conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.

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arXiv2021

Some Results on $k$-Critical $P_5$-Free Graphs

A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we prove that for every fixed integer $k\ge 1$, there are only finitely many $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3+P_2})$-free graphs. To prove the results we use a known structure theorem for ($P_5$,gem)-free graphs combined with properties of $k$-vertex-critical graphs. Moreover, we characterize all $k$-vertex-critical ($P_5$,gem)-free graphs and $(P_5,\overline{P_3+P_2})$-free graphs for $k \in \{4,5\}$ using a computer generation algorithm.

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arXiv2021

New bounds for Ramsey numbers $R(K_k-e,K_l-e)$

Let $R(H_1,H_2)$ denote the Ramsey number for the graphs $H_1, H_2$, and let $J_k$ be $K_k{-}e$. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: $49 \leq R(K_3,J_{12})$, $36 \leq R(J_4,K_8)$, $43 \leq R(J_4,J_{10})$, $52 \leq R(K_4,J_8)$, $37 \leq R(J_5,J_6)$, $43 \leq R(J_5,K_6)$, $65\leq R(J_5,J_7)$. We also use a gluing strategy to derive a new upper bound on $R(J_5,J_6)$. With both strategies combined, we prove the value of two Ramsey numbers: $R(J_5,J_6)=37$ and $R(J_5,J_7)=65$. We also show that the 64-vertex extremal Ramsey graph for $R(J_5,J_7)$ is unique. Furthermore, our algorithms also allow to establish new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: $R(W_7,W_4) = 21$, $R(W_7,W_7) = 19$, $R(K_{3,4},K_{3,4}) = 25$, and $R(K_{3,5}, K_{3,5})=33$.

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