# Jan Goedgebeur's research while affiliated with Ghent University and other places

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## Publications (67)

In 2012 we announced “the House of Graphs” (https://houseofgraphs.org) (Brinkmann et al. 2013), 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 grap...

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 Lov\'asz Local Lemma to extend Thomassen's in...

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (P6, diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We...

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...

The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders...

We propose a new heuristic algorithm for the Maximum Happy Vertices problem, 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 integrating a parameter $W$ that dictates the number of dynamic programming states to c...

Let R(H1,H2) denote the Ramsey number for the graphs H1,H2, and let Jk be Kk−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≤R(K3,J12), 36≤R(J4,K8), 43≤R(J4,J10), 52≤R(K4,J8), 37≤R(J5,J6), 43≤R(J5,K6), 6...

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (\(P_6\), diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (\(P_6\), diamond)-free graph G is no larger than the maximum of 6 and the clique number...

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 th...

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_...

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a ($P_6$, diamond)-free graph $G$ is no larger than the maximum of 6 and the clique number o...

The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders...

Objectives
Directed graph mapping (DGM) is a novel operator-independent automatic tool that can be applied to the identification of the atrial tachycardia (AT) mechanism. In the present study, for the first time, DGM was applied in complex AT cases, and diagnostic accuracy was evaluated.
Background
Catheter ablation of ATs still represents a chall...

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt be the path on t vertices. 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...

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 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on...

We present an algorithm to color a graph G with no triangle and no induced 7-vertex path (i.e., a {P7,C3}-free graph), where every vertex is assigned a list of possible colors which is a subset of {1,2,3}. While this is a special case of the problem solved in Bonomo et al. (2018) [1], that does not require the absence of triangles, the algorithm he...

It is well known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to make an efficient and practical implementation of an algorithm for the computatio...

Given two graphs and , a graph G is -free if it contains no induced subgraph isomorphic to or . Let be the path on t vertices. 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 algorithmi...

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular...

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. 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 gra...

Every $n$-vertex planar triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$. This was first conjectured by Albertson, Bollob\'as and Tucker, and was later proved by Heckman and Thomas. Fraughnaugh and Locke conjectured that the planarity requirement could be relaxed into just forbidding a few s...

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly reoccurring in numerous investigations of problems and conjectures surrounding snarks (connected cubic graphs...

It is well-known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex-set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to design a more efficient algorithm for the computation of the circular flow number...

Networks provide a powerful methodology with applications in a variety of biological, technological and social systems such as analysis of brain data, social networks, internet search engine algorithms, etc. To date, directed networks have not yet been applied to characterize the excitation of the human heart. In clinical practice, cardiac excitati...

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenverctor with no 0 elements. The problem of determining the orders $n$ for which $d$-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for $d \leq 4$. Here we solve the problem for all remaining cases $d\leq 11$ and det...

The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G. We present new results concerning the minimum leaf number of cubic graphs: we show that if G is a connected cubic graph of order n, then ml(G)≤n6+13, improving on the best known result in Salamon and Wiener (2008) and proving...

A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there
exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is
hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J.
Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here
we solve this problem. To this end...

We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number k ≥ 0 of hamiltonian cycles, which is especially efficient for small k. Our main findings, combining applications of this algorithm and existing algorithms with new theoretical results, revolve around graphs containing exactly one hamiltonian cycle (...

Networks provide a powerful methodology for a variety of biological, technological and social systems, such as analysis of brain data, social networks, internet search machines, etc. Interestingly, directed networks were not yet applied to characterize the excitation of the human heart, although this system would perfect fit for this type of analys...

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at
least $g$. We consider the problem of determining $n_g(k)$ for small values of
$k$ and $g$. After giving an overview of what is known about $n_g(k)$, we
provide some new lower bounds based on exhaustive searches, and then obtain
several new upper bounds using computer algor...

The oddness of a cubic graph is the smallest number of odd circuits in a 2-factor of the graph. This invariant is widely considered to be one of the most important measures of uncolourability of cubic graphs and as such has been repeatedly reoccurring in numerous investigations of problems and conjectures surrounding snarks (connected cubic graphs...

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenvector with no zero elements. The problem of determining the orders n for which d-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d ≤ 4. Here we solve the problem for all remaining cases d ≤ 11 and determine the...

We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number $k \ge 0$ of hamiltonian cycles, which is especially efficient for small $k$. Our main findings, combining applications of this algorithm and existing algorithms with new theoretical results, revolve around graphs containing exactly one hamiltonian c...

The \emph{minimum leaf number} $\hbox{ml} (G)$ of a connected graph $G$ is defined as the minimum number of leaves of the spanning trees of $G$. We present new results concerning the minimum leaf number of cubic graphs: we show that if $G$ is a connected cubic graph of order $n$, then $\mathrm{ml}(G) \leq \frac{n}6 + \frac13$, improving on the best...

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algor...

The well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5...

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of appr...

A platypus graph is a non-hamiltonian graph for which every vertex-deleted subgraph is traceable. They are closely related to families of graphs satisfying interesting conditions regarding longest paths and longest cycles, for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian graphs.
In this paper, we first investigate cubic plat...

A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J. Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and remarked and that the...

A nut graph is a graph on at least 2 vertices whose adjacency matrix has nullity 1 and for which non-trivial kernel vectors do not contain a zero. Chemical graphs are connected, with maximum degree at most three. We present a new algorithm for the exhaustive generation of non-isomorphic nut graphs. Using this algorithm, we determined all nut graphs...

A graph G is hypohamiltonian if G is non-hamiltonian and for every vertex v in G, the graph G−v is hamiltonian. McKay asked in [J. Graph Theory 85 (2017) 7–11] whether infinitely many planar cubic hypohamiltonian graphs of girth 5 exist. We settle this question affirmatively.

A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 23 vertices and all triangle-free 5-chromatic graphs on 24 vertices with maximum...

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except...

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph ad...

We characterize all graphs $H$ for which there are only finitely many $H$-free 4-vertex-critical graphs. Such a characterization was known only in the case when $H$ is connected. This solves a problem posed by Golovach et al. As a second result, we characterize all graphs $H$ for which there are only finitely many $H$-free minimal obstructions for...

We discuss an omission in the statement and proof of Fiorini's 1983 theorem on hypohamiltonian snarks and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen showed that hypohamiltonian snarks exist for some $n \ge 10$ and each even $n \ge 92$. We rectify Steffen's proof by providing a...

We describe an algorithm for generating all k-critical \({\mathcal H}\)-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many 4-critical \((P_7,C_k)\)-free graphs, for both \(k=4\) and \(k=5\). We also show that there are only finitely many 4-critical \((P_8,C_4)\)-free graphs. For each case...

A graph $G$ is hypohamiltonian if $G$ is non-hamiltonian and $G - v$ is hamiltonian for every $v \in V(G)$. In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove t...

We describe two new algorithms for the generation of all non-isomorphic cubic
graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$
and show how these algorithms can be efficiently restricted to generate snarks
with girth at least $k$.
Our implementation of these algorithms is more than 30, respectively 40 times
faster tha...

For each $d>0$, we find all the smallest fullerenes for which the least
distance between two pentagons is $d$. We also show that for each $d$ there is
an $h_d$ such that fullerenes with pentagons at least distance $d$ apart and
any number of hexagons greater than or equal to $h_d$ exist.
We also determine the number of fullerenes where the minimum...

We describe an algorithm for generating all $k$-critical $\mathcal H$-free
graphs, based on a method of Ho\`{a}ng et al. Using this algorithm, we prove
that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for
both $k=4$ and $k=5$. We also show that there are only finitely many
$4$-critical graphs $(P_8,C_4)$-free graphs. For each...

We prove that there are 24 4-critical $P_6$-free graphs, and give the
complete list. We remark that, if $H$ is connected and not a subgraph of $P_6$,
there are infinitely many 4-critical $H$-free graphs. Our result answers
questions of Golovach et al. and Seymour.

We describe a new construction algorithm for the recursive generation of all
non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm
stays entirely within the class of IPR fullerenes, that is: every IPR fullerene
is constructed by expanding a smaller IPR fullerene unless it belongs to
limited class of irreducible IPR fullerenes...

A graph $G$ is pseudo 2-factor isomorphic if the parity of the number of
cycles in a 2-factor is the same for all 2-factors of $G$. Abreu et al.
conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only
essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs
(Abreu et al., Journal of Combinatorial Theory...

The latest transportation systems require the best routes in a large network with respect to multiple objectives simultaneously to be calculated in a very short time. The label setting algorithm of Martins efficiently finds this set of Pareto optimal paths, but sometimes tends to be slow, especially for large networks such as transportation network...

Using computer algorithms we establish that the Ramsey number R(3, K10 - e) is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of R(3,Kk - e) for 11 ≤ k ≤ 16, and show by construction a new lower bound 55 ≤ R(3, K13 - e). The new upper bounds on R(3,Kk - e) are obtained...

Using computer algorithms we establish that the Ramsey number R(3,K 10 -e) is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of R(3,K k -e) for 11≤k≤16, and show by construction a new lower bound 55≤R(3,K 13 -e).The new upper bounds on R(3,K k -e) are obtained by using...

Using computational techniques we derive six new upper bounds on the
classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <=
68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are
improvements by one over the previously best known bounds.
Let e(3,k,n) denote the minimum number of edges in any triangle-free graph o...

We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes - fullgen - and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for...

In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12
005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for
which we could not determine the Ramsey number. Most likely these graphs need
approaches focusing on each individual graph in order to determine their
triangle Ramsey number. The results were...

For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementat...

In this note we present House of Graphs (http://hog.grinvin.org) which is a
new database of graphs. The key principle is to have a searchable database and
offer -- next to complete lists of some graph classes -- also a list of special
graphs that already turned out to be interesting and relevant in the study of
graph theoretic problems or as counte...

In this note, we give the result of a computer search for the smallest fullerene that does not allow a face spiral code as used by Manolopoulos and Fowler and adopted in IUPAC recommendations for fullerene nomenclature. The search enumerated all the small fullerenes on up to 400 vertices and the conclusion is that the smallest fullerene without a f...

Discrete Algorithms
International audience
We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5.

As traffic routing applications usually are heavily burdened due to the many requests, a low execution time of the shortest path algorithms is of uttermost importance. In this article two goal-directed heuristics are presented, which reduce this execution time. By guiding the search toward the destination and neglecting the less interesting areas o...

## Citations

... In 2021, Cameron et al. [2] improved the χ-bounding function of (P 6 , diamond)-free graphs to ω(G) + 3. In a recent paper [8], Goedgebeur et ...

... If a graph G is (P 6 , C 4 )-free or (P 5 , P 4 ∨ K 1 )-free or (P 4 + K 1 , P 4 ∨ K 1 )free, then G satisfies χ(G) ≤ ⌈ 5ω(G) 4 ⌉; see [6,16,17]. Every (P 6 , diamond)-free graph G satisfies χ(G) ≤ max{6, ω(G)} [14]. ...

Reference: Coloring ($P_5$, kite)-free graphs

... The complete lists of all k-regular graphs on n vertices containing h n (k) or h 2 n (k) hamiltonian cycles-whose counts are mentioned between parentheses in Table 1-can be obtained from the database of interesting graphs from the House of Graphs [3] by searching for the keywords "regular * minimum number of hamiltonian cycles" to allow other researchers to inspect or independently verify these results. ...

... Garcia et al. [9] in 2019 provides a natural bijection between the associated graphs of generalized crowns and a particular family of block circulant graphs. Recently in 2022 Goedgebeur and Van Overberghe [10] presented algorithms to establish the new lower bounds and exact values on Ramsey numbers involving circulant and block-circulant graphs. ...

... DGM has previously been investigated in the setting of atrial tachycardia with excellent results but has not yet been applied to VT circuits. 4,5 A network, in the most general sense, is a collection of nodes connected by links, which can represent a diverse system. Network theory has broader applications in biology and social sciences from modeling of molecules, social networks, and the spread of diseases. ...

... The reason for finiteness is that if we know there are only finitely many k-vertex-critical graphs, then there is a polynomial-time algorithm for (k − 1)-coloring graphs in that class. In 2021, Kameron, Goedgebeur, Huang and Shi [4] obtained the following dichotomy result for k-vertex-critical (P 5 , H)-free graphs when |H| = 4. ...

Reference: Vertex-Critical $(P_5, chair)$-Free Graphs

... There have also been computational efforts concerning Conjecture 1.1. The first of which, by Holton et al. [HMM85] in 1985, also introduced an elegant generation method for the graphs in B and this is still the method used in recent efforts (see [BGM21]). To support the relevance of Lemma 1.2 to the effort of resolving Conjecture 1.1, we provide a short list of amendments to the generation procedure in Section 5 that allow us to keep track of whether a graph we have generated is a brace or not. ...

Reference: Matching theory and Barnette's conjecture

... This type of algorithm is used when the problem is NP-hard and when the input data is large (in terms of the number of vertices and edges in a graph). In addition, there are other algorithms for the graph vertex coloring problem [29], [30]. ...

... As well, there are only finitely many 6-vertex-critical (P 5 , banner)-free graphs [1]. In recent work, it was shown for k ≥ 5 that there is a finite number of k-vertex-critical (P 5 , H)-free graphs, where H has four vertices and H is neither 2K 2 nor K 3 + P 1 [2]. One of the more significant recent result is the following dichotomy theorem: ...

... An effective and widely used approach to prove theorems on subcubic graphs is to take a hypothetical minimum counterexample and then analyse small cut sets as well as the local structure around vertices of small degree, followed by an analysis of the local structure in a highly connected cubic graph, eventually leading to a contradiction (see e.g. [4,3,21] for some recent examples). This paper is no exception. ...

Reference: Minimum Maximal Matchings in Cubic Graphs