arXiv subjects

Stijn Cambie

Publications and source records attributed to Stijn Cambie.

At least 37 records · Page 2Linked to original sources

arXiv2025

On edge-colouring-games by Erdős, and Bensmail and Mc Inerney

We study two games proposed by Erdős, and one game by Bensmail and Mc Inerney, all sharing a common setup: two players alternately colour edges of a complete graph, or in the biased version, they colour $p$ and $q$ edges respectively on their turns, aiming to maximise a graph parameter determined by their respective induced subgraphs. In the unbiased case, we give a first reduction towards confirming the conjecture of Bensmail and Mc Inerney, propose a conjecture for Erdős' game on maximum degree, and extend the clique and maximum-degree versions to edge-transitive and regular graphs. In the biased case, the maximum-degree and vertex-capturing games are resolved, and we prove the clique game with $(p,q)=(1,3)$.

math.CO↗

arXiv2025

Note on extremal problems about connected subgraph sums

For a graph $G$ with vertex assignment $c:V(G)\to \mathbb{Z}^+$, we define $\sum_{v\in V(H)}c(v)$ for $H$ a connected subgraph of $G$ as a connected subgraph sum of $G$. We study the set $S(G,c)$ of connected subgraph sums and, in particular, resolve a problem posed by Solomon Lo in a strong form. We show that for each $n$-vertex graph, there is a vertex assignment $c:V(G)\to \{1,\dots,12n^2\}$ such that for every $n$-vertex graph $G'\not\cong G$ and vertex assignment $c'$ for $G'$, the corresponding collections of connected subgraph sums are different (i.e., $S(G,c)\neq S(G',c')$). We also provide some remarks on vertex assignments of a graph $G$ for which all connected subgraph sums are different.

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arXiv2025

Uniform Šoltés' hypergraphs and Šoltés' weighted graphs

A Šoltés' hypergraph is a hypergraph for which the removal of any of its vertices does not change its total distance. We prove that every uniform Šoltés' hypergraph has order at least $10$, there exist uniform Šoltés' hypergraphs for almost every order or uniformity, and there exist a non-regular uniform Šoltés' hypergraph. By also providing infinitely many weighted Šoltés' graphs, we conclude that Šoltés' problem can be answered positively for the most natural generalisations of graphs.

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arXiv2025

Reconfiguration of List Colourings

Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|\ge \mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $Δ$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $\mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to $\{1,\ldots,Δ+1\}$.

math.CO↗

arXiv2025

On the order of intersecting hypergraphs

Determining the maximum number of edges in an intersecting hypergraph on a fixed ground set under additional constraints is one of the central topics in extremal combinatorics. In contrast, there are few results on analogous problems concerning the maximum order of such hypergraphs. In this paper, we systematically study these vertex analogues.

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arXiv2025

On Erdős problem $\# 648$

We establish the order of the maximum length of an increasing sequence, bounded by $n$, in which the largest prime divisor of the elements form a decreasing sequence.

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arXiv2025

Trees maximizing the number of almost-perfect matchings

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.

math.CO↗

arXiv2025

Better bounds for the union-closed sets conjecture using the entropy approach

We improve the best known constant $\frac{3-\sqrt 5}{2}$ for which the union-closed conjecture is known to be true, by using dependent samples as suggested by Sawin and the entropy approach on this problem initiated by Gilmer. Meanwhile, we focus on the intuition behind this entropy approach and its boundaries.

math.CO↗

arXiv2025

Sharp results for the Erdős, Pach, Pollack and Tuza problem

We consider the Erdős, Pach, Pollack and Tuza problem, asking for the maximum diameter of a graph with given order $n$, minimum degree $δ$ and clique number at most $ω$. We solve their problem asymptotically for the first hard case, $ω\leq 3$, for the smallest values of $δ$ by determining the smallest rational number $f(δ)$ such that $diam(G) \leq f(δ)n+O(1)$ for all graphs $G$ with order $n$, minimum degree $δ$ and clique number $ω\leq 3$. We also consider the weaker version where the clique number $ω\leq 3$ is replaced by having chromatic number $χ\leq 3$ and solve this version for small $δ$, thereby yielding a counterexample to a conjecture of Erdős et al. in a regime where this conjecture was still open. When restricting the conjecture to graphs with chromatic number $χ\leq 3$, we show that this counterexample appears for the smallest possible $δ$, namely $δ=16.$

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arXiv2025

Resolution of Erdős' problems about unimodularity

Letting $δ_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erdős wondered if $δ_1(n,m)$ is unimodular for fixed $n$. We prove this is false in general, as the sequence $(δ_1(n,m))$ has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; $n = 1$. We also solve the question on unimodality of the density of integers whose $k^{th}$ prime is $p$.

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arXiv2025

The average solution of a TSP instance in a graph

We define the average $k$-TSP distance $μ_{tsp,k}$ of a graph $G$ as the average length of a shortest walk visiting $k$ vertices, i.e. the expected length of the solution for a random TSP instance with $k$ uniformly random chosen vertices. We prove relations with the average $k$-Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for $μ_{tsp,k}(G)$ given the order of the graph.

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arXiv2024

Abundancy of $z$-\v Soltés' digraphs

We prove the existence of infinitely many \v Soltés' digraphs, the digraph analogue of \v Soltés' graphs. We also give an example of a \v Soltés' digraph with trivial automorphism group.

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arXiv2024

Sharp Bounds on Lengths of Linear Recolouring Sequences

A recolouring sequence, between $k$-colourings $α$ and $β$ of a graph $G$, transforms $α$ into $β$ by recolouring one vertex at a time, such that after each recolouring step we again have a proper $k$-colouring of $G$. The diameter of the $k$-recolouring graph, $\textrm{diam}~\mathcal{C}_k(G)$, is the maximum over all pairs $α$ and $β$ of the minimum length of a recolouring sequence from $α$ to $β$. Much previous work has focused on determining the asymptotics of $\textrm{diam}~\mathcal{C}_k(G)$: Is it $Θ(|G|)$? Is it $Θ(|G|^2)$? Or even larger? Here we focus on graphs for which $\textrm{diam}~\mathcal{C}_k(G)=Θ(|G|)$, and seek to determine more precisely the multiplicative constant implicit in the $Θ()$. In particular, for each $k\ge 3$, for all positive integers $p$ and $q$ we exactly determine $\textrm{diam}~\mathcal{C}_k(K_{p,q})$, up to a small additive constant. We also sharpen a recolouring lemma that has been used in multiple papers, proving an optimal version. This improves the multiplicative constant in various prior results. Finally, we investigate plausible relationships between similar reconfiguration graphs.

math.CO↗

arXiv2024

On the main distance-based entropies: the eccentricity- and Wiener-entropy

We define the Wiener-entropy, which is together with the eccentricity-entropy one of the most natural distance-based graph entropies. By deriving the (asymptotic) extremal behaviour, we conclude that the Wiener-entropy of graphs of a given order is more spread than is the case for the eccentricity-entropy. We solve $3$ conjectures on the eccentricity-entropy and give a conjecture on the Wiener-entropy related to some surprising behaviour on the graph minimizing it.

math.CO↗

arXiv2024

Packing colourings in complete bipartite graphs and the inverse problem for correspondence packing

Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing. However, even for complete bipartite graphs, the list chromatic number can be arbitrarily large, and its exact determination is generally difficult. For the packing variant, this question becomes even harder. In this paper, we study the correspondence- and list packing numbers of (asymmetric) complete bipartite graphs. In the most asymmetric cases, Latin squares come into play. Our results show that every $z \in \mathbb Z^+ \setminus {3}$ can be equal to the correspondence packing number of a graph. Additionally, we disprove a recent conjecture that relates the list packing number and the list flexibility number.

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arXiv2024

Open problems of the 32nd 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 32nd edition of the workshop which took place in Poprad, Slovakia during September 8-13, 2024 (see the workshop webpage https://candc.upjs.sk).

math.CO↗

arXiv2024

Resolution of an Erdős' problem on least common multiples

Erdős posed the question whether there exist infinitely many sets of consecutive numbers whose least common multiple (lcm) exceeds the lcm of another, larger set with greater consecutive numbers. In this paper, we answer this question affirmatively by proving that the ratio of the lcm's can be made arbitrarily large.

math.GM↗

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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