Searcharxiv⌕ Search

arXiv subjects

Stijn Cambie

Publications and source records attributed to Stijn Cambie.

At least 55 records · Page 3Linked to original sources
arXiv2024

The minimum number of maximal independent sets in twin-free graphs

The problem of determining the maximum number of maximal independent sets in certain graph classes dates back to a paper of Miller and Muller and a question of Erdős and Moser from the 1960s. The minimum was always considered to be less interesting due to simple examples such as stars. In this paper we show that the problem becomes interesting when restricted to twin-free graphs, where no two vertices have the same open neighbourhood. We consider the question for arbitrary graphs, bipartite graphs and trees. The minimum number of maximal independent sets turns out to be logarithmic in the number of vertices for arbitrary graphs, linear for bipartite graphs and exponential for trees. In the latter case, the minimum and the extremal graphs have been determined earlier by Taletski\uı and Malyshev, but we present a shorter proof.

math.CO↗
arXiv2024

Hypergraph saturation for the bow tie

Erdős and Sós initiated the study of the maximum size of a $k$-uniform set system, for $k \geq 4$, with no singleton intersections $50$ years ago. In this work, we investigate the dual problem: finding the minimum size of a $k$-uniform hypergraph with no singleton intersections, such that adding any missing hyperedge forces a singleton intersection. These problems, known as saturation and semi-saturation, are typically challenging. Our focus is on an elementary-to-state case in the line of work by Erdős, Füredi and Tuza. We establish tight linear bounds for $k=4$, marking one of the first non-obvious cases with such a bound.

math.CO↗
arXiv2024

Descents and inversions in powers of permutations

In this paper, we generalise several recent results by Archer and Geary on descents in powers of permutations, and confirm all their conjectures. Specifically, for all $k\in\mathbb{Z}^+$, we prove explicit formulas for the expected numbers of descents and inversions in the $k$-th powers of permutations in $\mathcal{S}_n$ for all $n\geq2k+1$. We also compute the number of Grassmanian permutations in $\mathcal{S}_n$ whose $k$-th powers remain Grassmanian, and the number of permutations in $\mathcal{S}_n$ whose $k$-th powers have the maximum number of descents.

math.CO↗
arXiv2024

Towards the essence of Šoltés' problem

We explore the question asking for graphs $G$ for which the total distance decreases, possibly by a fixed constant $k$, upon the removal of any of its vertices. We obtain results leading to intuition and doubts for the Šoltés' problem ($k=0$) and its conjectures.

math.CO↗
arXiv2024

Šoltés' hypergraphs

More than $30$ years ago, Šoltés observed that the total distance of the graph $C_{11}$ does not change by deleting a vertex, and wondered about the existence of other such graphs, called Šoltés graphs. We extend the definition of Šoltés' graphs to Šoltés' hypergraphs, determine all orders for which a Šoltés' hypergraph exists, observe infinitely many uniform Šoltés' hypergraphs, and find the Šoltés' hypergraph with minimum size (spoiler: it is not $C_{11}$).

math.CO↗
arXiv2024

A precise condition for independent transversals in bipartite covers

Given a bipartite graph $H=(V=V_A\cup V_B,E)$ in which any vertex in $V_A$ (resp.~$V_B$) has degree at most $D_A$ (resp.~$D_B$), suppose there is a partition of $V$ that is a refinement of the bipartition $V_A\cup V_B$ such that the parts in $V_A$ (resp.~$V_B$) have size at least $k_A$ (resp.~$k_B$). We prove that the condition $D_A/k_B+D_B/k_A\le 1$ is sufficient for the existence of an independent set of vertices of $H$ that is simultaneously transversal to the partition, and show moreover that this condition is sharp. This result is a bipartite refinement of two well-known results on independent transversals, one due to the second author and the other due to Szabó and Tardos.

math.CO↗
arXiv2023

Disjoint list-colorings for planar graphs

One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find $3$ disjoint $L$-colorings (a packing), or $2$ disjoint $L$-colorings, or a collection of $L$-colorings that to every vertex assigns every color on average in one third of the cases (a fractional packing)? We prove that the packing is impossible, but two disjoint $L$-colorings are guaranteed if the girth is at least $8$, and a fractional packing exists when the girth is at least $6.$ For a graph $G$, the least $k$ such that there are always $k$ disjoint proper list-colorings whenever we have lists all of size $k$ associated to the vertices is called the list packing number of $G$. We lower the two-times-degeneracy upper bound for the list packing number of planar graphs of girth $3,4$ or $5$. As immediate corollaries, we improve bounds for $ε$-flexibility of classes of planar graphs with a given girth. For instance, where previously Dvořák et al. proved that planar graphs of girth $6$ are (weighted) $ε$-flexibly $3$-choosable for an extremely small value of $ε$, we obtain the optimal value $ε=\frac{1}{3}$. Finally, we completely determine and show interesting behavior on the packing numbers for $H$-minor-free graphs for some small graphs $H.$

math.CO↗
arXiv2023

Counterexamples to conjectures on the occupancy fraction of graphs

The occupancy fraction of a graph is a (normalized) measure on the size of independent sets under the hard-core model, depending on a variable (fugacity) $λ.$ We present a criterion for finding the graph with minimum occupancy fraction among graphs with a fixed order, and disprove five conjectures on the extremes of the occupancy fraction and (normalized) independence polynomial for certain graph classes of regular graphs with a given girth.

math.CO↗
arXiv2023

Optimally Reconfiguring List and Correspondence Colourings

The reconfiguration graph $\mathcal{C}_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $\mathcal{C}_k(G)$ are adjacent precisely when those $k$-colourings differ on a single vertex of $G$. Much work has focused on bounding the maximum value of ${\rm{diam}}~\mathcal{C}_k(G)$ over all $n$-vertex graphs $G$. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if $L$ is a list-assignment for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+μ(G)$. We also conjecture that if $(L,H)$ is a correspondence cover for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+τ(G)$. (Here $μ(G)$ and $τ(G)$ denote the matching number and vertex cover number of $G$.) For every graph $G$, we give constructions showing that both conjectures are best possible. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+2μ(G)$ and ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+2τ(G)$. Our second main result proves that both conjectured bounds hold, whenever all $v$ satisfy $|L(v)|\ge 2d(v)+1$. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree.

math.CO↗
arXiv2023

Bounding mean orders of sub-$k$-trees of $k$-trees

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.

math.CO↗
arXiv2023

Corrigendum on Wiener index, Zagreb Indices and Harary index of Eulerian graphs

In the original article ``Wiener index of Eulerian graphs'' [Discrete Applied Mathematics Volume 162, 10 January 2014, Pages 247-250], the authors state that the Wiener index (total distance) of an Eulerian graph is maximized by the cycle. We explain that the initial proof contains a flaw and note that it is a corollary of a result by Plesník, since an Eulerian graph is $2$-edge-connected. The same incorrect proof is used in two referencing papers, ``Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs'' [Bull. Malays. Math. Sci. Soc. (2019) 42:67-78] and ``Harary index of Eulerian graphs'' [J. Math. Chem., 59(5):1378-1394, 2021]. We give proofs of the main results of those papers and the $2$-edge-connected analogues.

math.CO↗
arXiv2023

Decreasing the mean subtree order by adding $k$ edges

The mean subtree order of a given graph $G$, denoted $μ(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $μ(H) > μ(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $μ(H) < μ(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $μ(H) < μ(G)$. Furthermore, they proposed that $μ(K_m+nK_1) < μ(K_{m, n})$ provided $n\gg m$. In this note, we confirm these two conjectures.

math.CO↗
arXiv2023

The Erdős distinct subset sums problem in a modular setting

We prove the following variant of the Erdős distinct subset sums problem. Given $t \ge 0$ and sufficiently large $n$, every $n$-element set $A$ whose subset sums are distinct modulo $N=2^n+t$ satisfies $$\max A \ge \Big(\frac{1}{3}-o(1)\Big)N.$$ Furthermore, we provide examples showing that the constant $\frac 13$ is best possible. For small values of $t$, we characterise the structure of all sumset-distinct sets modulo $N=2^n+t$ of cardinality $n$.

math.CO↗
arXiv2023

When removing an independent set is optimal for reducing the chromatic number

How large must the chromatic number of a graph be, in terms of the graph's maximum degree, to ensure that the most efficient way to reduce the chromatic number by removing vertices is to remove an independent set? By a reduction to a powerful, known stability form of Brooks' theorem, we answer this question precisely, determining the threshold to within two values (and indeed sometimes a unique value) for graphs of sufficiently large maximum degree.

math.CO↗
arXiv2023

Progress on the union-closed conjecture and offsprings in winter 2022-2023

Mathematicians had little idea whether the easy-to-state union-closed conjecture was true or false even after $40$ years. However, last winter saw a surge of interest in the conjecture and its variants, initiated by the contribution of a researcher at Google. Justin Gilmer [arXiv:2211.09055] made a significant breakthrough by discovering a first constant lower bound for the proportion of the most common element in a union-closed family.

math.CO↗
arXiv2023

A proof of Frankl's conjecture on cross-union families

The families $\mathcal F_0,\ldots,\mathcal F_s$ of $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ are called cross-union if there is no choice of $F_0\in \mathcal F_0, \ldots, F_s\in \mathcal F_s$ such that $F_0\cup\ldots\cup F_s=[n]$. A natural generalization of the celebrated Erdős--Ko--Rado theorem, due to Frankl and Tokushige, states that for $n\le (s+1)k$ the geometric mean of $\lvert \mathcal F_i\rvert$ is at most $\binom{n-1}{k}$. Frankl conjectured that the same should hold for the arithmetic mean under some mild conditions. We prove Frankl's conjecture in a strong form by showing that the unique (up to isomorphism) maximizer for the arithmetic mean of cross-union families is the natural one $\mathcal F_0=\ldots=\mathcal F_s={[n-1]\choose k}$.

math.CO↗
arXiv2023

Extremal Binary PFAs with Small Number of States

The largest known reset thresholds for DFAs are equal to $(n-1)^2$, where $n$ is the number of states. This is conjectured to be the maximum possible. PFAs (with partial transition function) can have exponentially large reset thresholds. This is still true if we restrict to binary PFAs. However, asymptotics do not give conclusions for fixed $n$. We prove that the maximal reset threshold for binary PFAs is strictly greater than $(n-1)^2$ if and only if $n\geq 6$. These results are mostly based on the analysis of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-known Černý automata; for $n\leq 10$ it contains a binary PFA with maximal possible reset threshold; for all $n\geq 6$ it contains a PFA with reset threshold larger than the maximum known for DFAs. Analysis of this family reveals remarkable patterns involving the Fibonacci numbers and related sequences such as the Padovan sequence. We derive explicit formulas for the reset thresholds in terms of these recurrent sequences. Asymptotically the Černý family gives reset thresholds of polynomial order. We prove that PFAs in the family are not extremal for $n\geq 41$. For that purpose, we present an improvement of Martyugin's prime number construction of binary PFAs.

cs.FL↗
arXiv2023

Packing list-colourings

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$-list-assignment $L$ of a graph $G$, which is the assignment of a list $L(v)$ of $k$ colours to each vertex $v\in V(G)$, we study the existence of $k$ pairwise-disjoint proper colourings of $G$ using colours from these lists. We may refer to this as a \emph{list-packing}. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest $k$ for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of $G$. (The reader might already find it interesting that such a minimal $k$ is well defined.) We also pursue a more focused study of the case when $G$ is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal $k$ above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.

math.CO↗