arXiv subjects

Stijn Cambie

Publications and source records attributed to Stijn Cambie.

At least 73 records · Page 4Linked to original sources

arXiv2023

List packing number of bounded degree graphs

We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list packing number contrasts with that of the list chromatic number, particularly in the context of bounded degree graphs. The main question we pursue is whether every graph with maximum degree $Δ$ has list packing number at most $Δ+1$. Our results highlight the subtleties of list packing and the barriers to, for example, pursuing a Brooks'-type theorem for the list packing number.

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arXiv2023

The maximum Wiener index of a uniform hypergraph

The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected $n$-vertex $k$-uniform hypergraph and characterize for every~$n$ all hypergraphs attaining the maximum Wiener index.

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arXiv2023

Many Hamiltonian subsets in large graphs with given density

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ($\approx 2^{d+1}$). Among others, our bound implies that an $n$-vertex $C_4$-free graphs with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.

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arXiv2022

Set systems without a simplex, Helly hypergraphs and union-efficient families

We present equivalent formulations for concepts related to set families for which every subfamily with empty intersection has a bounded sub-collection with empty intersection. Hereby, we summarize the progress on the related questions about the maximum size of such families. In this work we solve a boundary case of a problem of Tuza for non-trivial $q$-Helly families, by applying Karamata's inequality and determining the minimum size of a $2$-self-centered graph for which the common neighborhood of every pair of vertices contains a clique of size $q-2$.

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arXiv2022

Extremal and monotone behaviour of the Sudoku number and related critical set parameters

The Sudoku number has been defined under various names, indicating it is a natural concept. There are four variants of this parameter, that can be related to the maximum and minimum size of a critical set in a graph colouring problem. For each of these four related parameters, we present some simple characterizations of the graphs attaining the maximum possible values. As a main result, we answer a question by Cooper and Kirkpatrick, showing that there is monotone behaviour in the number of colours for only two of the four parameters. We investigate the monotone behaviour for the subgraph-order as well. For Latin squares and the Sudoku, we solve some variants for hypergraph colouring.

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arXiv2022

On the relationship between variable Wiener index and variable Szeged index

We resolve two conjectures of Hriňáková, Knor and Škrekovski (2019) concerning the relationship between the variable Wiener index and variable Szeged index for a connected, non-complete graph, one of which would imply the other. The strong conjecture is that for any such graph there is a critical exponent in $(0,1]$, below which the variable Wiener index is larger and above which the variable Szeged index is larger. The weak conjecture is that the variable Szeged index is always larger for any exponent exceeding $1$. They proved the weak conjecture for bipartite graphs, and the strong conjecture for trees. In this note we disprove the strong conjecture, although we show that it is true for almost all graphs, and for bipartite and block graphs. We also show that the weak conjecture holds for all graphs by proving a majorization relationship.

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arXiv2022

Extremal values of degree-based entropies of bipartite graphs

We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Young tableaux, which make it easier to compare the entropy of related graphs. We conclude that the general characterization of the extremal graphs is a difficult problem, due to its connections with number theory, but they are easy to find for specific values of the order $n$ and size $m$. We also give a direct argument to characterize the graphs maximizing the entropy given order and size. We indicate that some of our ideas extend to other degree-based topological indices as well.

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arXiv2022

Extremal entropy for graphs with given size

The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal values given natural constraints. In this paper, we prove that the graphs with given size that minimize the first degree-based entropy are the colex graphs.

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arXiv2022

Resolution of Yan's conjecture on entropy of graphs

The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. In this paper, we characterize the connected graphs with given order $n$ and size $m$ that minimize the first degree-based entropy whenever $n-1 \le m \le 2n-3,$ thus extending and proving a conjecture by Yan.

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arXiv2022

Maximum size of digraphs of given radius

In $1967$, Vizing determined the maximum size of a graph with given order and radius. In $1973$, Fridman answered the same question for digraphs with given order and outradius. We investigate that question when restricting to biconnected digraphs. Biconnected digraphs are the digraphs with a finite total distance and hence the interesting ones, as we want to note a connection between minimizing the total distance and maximizing the size under the same constraints. We characterize the extremal digraphs maximizing the size among all biconnected digraphs of order $n$ and outradius $3$, as well as when the order is sufficiently large compared to the outradius. As such, we solve a problem of Dankelmann asymptotically. We also consider these questions for bipartite digraphs and solve a second problem of Dankelmann partially.

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arXiv2022

Extremal total distance of graphs of given radius I

In 1984, Plesník determined the minimum total distance for given order and diameter and characterized the extremal graphs and digraphs. We prove the analog for given order and radius, when the order is sufficiently large compared to the radius. This confirms asymptotically a conjecture of Chen et al. We also state an analog of the conjecture of Chen et al for digraphs and prove it for sufficiently large order.

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arXiv2021

Maximising line subgraphs of diameter at most $t$

We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree $Δ$ with more than $1.5 Δ^t$ edges, its line graph must have diameter larger than $t$. In the case where the graph contains no cycle of length $2t+1$, we can improve the bound on the number of edges to one that is exact for $t\in\{1,2,3,4,6\}$. In the case $Δ=3$ and $t=3$, we obtain an exact bound. Our results also have implications for the related problem of bounding the distance-$t$ chromatic index, $t>2$; in particular, for this we obtain an upper bound of $1.941Δ^t$ for graphs of large enough maximum degree $Δ$, markedly improving upon earlier bounds for this parameter.

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arXiv2021

Independent transversals in bipartite correspondence-covers

Suppose $G$ and $H$ are bipartite graphs and $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph of $H$ induced between $L(v)$ and $L(v')$ is a matching whenever $vv'\in E(G)$. We show for each $\varepsilon>0$ that, if $H$ has maximum degree $D$ and $|L(v)| \ge (1+\varepsilon)D/\log D$ for all $v\in V(G)$, then $H$ admits an independent transversal with respect to $L$, provided $D$ is sufficiently large. This bound on the part sizes is asymptotically sharp up to a factor $2$. We also show some asymmetric variants of this result.

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arXiv2021

Five results on maximizing topological indices in graphs

In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.

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arXiv2021

Asymmetric list sizes in bipartite graphs

Given a bipartite graph with parts $A$ and $B$ having maximum degrees at most $Δ_A$ and $Δ_B$, respectively, consider a list assignment such that every vertex in $A$ or $B$ is given a list of colours of size $k_A$ or $k_B$, respectively. We prove some general sufficient conditions in terms of $Δ_A$, $Δ_B$, $k_A$, $k_B$ to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where $Δ_A=Δ_B=Δ$, $k_A=\log Δ$ and $k_B=(1+o(1))Δ/\logΔ$ as $Δ\to\infty$. This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteed * if $k_A \ge Δ_A^\varepsilon$ and $k_B \ge Δ_B^\varepsilon$ for any $\varepsilon>0$ provided $Δ_A$ and $Δ_B$ are large enough; * if $k_A \ge C \logΔ_B$ and $k_B \ge C \logΔ_A$ for some absolute constant $C>1$; or * if $Δ_A=Δ_B = Δ$ and $ k_B \ge C (Δ/\logΔ)^{1/k_A}\log Δ$ for some absolute constant $C>0$. These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures.

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arXiv2021

Hadwiger's conjecture implies a conjecture of Füredi-Gyárfás-Simonyi

One of the most important open problems in the field of graph colouring or even graph theory is the conjecture of Hadwiger. This conjecture was the inspiration for many mathematical works, one of them being the work of Füredi, Gyárfás and Simonyi in which they "risked" to conjecture the precise bound for a graph with independence number $2$ to contain a certain connected matching. We prove that their conjecture would be a corollary of Hadwiger's conjecture or equivalently if their risky conjecture would be false, then Hadwiger's conjecture would be false as well.

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arXiv2021

On the maximum mean subtree order of trees

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tree, is always attained by some caterpillar. While we do not completely resolve this conjecture, we find some evidence in its favor by proving different features of trees that attain the maximum. For example, we show that the diameter of a tree of order $n$ with maximum mean subtree order must be very close to $n$. Moreover, we show that the maximum mean subtree order is equal to $n - 2\log_2 n + O(1)$. For the local mean subtree order, which is the average order of all subtrees containing a fixed vertex, we can be even more precise: we show that its maximum is always attained by a broom and that it is equal to $n - \log_2 n + O(1)$.

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arXiv2021

Punctured intervals tile $\mathbb Z^3$

Extending the methods of Metrebian (2018), we prove that any symmetric punctured interval tiles $\mathbb Z^3$. This solves two questions of Metrebian and completely resolves a question of Gruslys, Leader and Tan. We also pose a question that asks whether there is a relation between the genus $g$ (number of holes) in a one-dimensional tile $T$ and a uniform bound $d$ such that $T$ tiles $\mathbb{Z}^d$. An affirmative answer would generalize a conjecture of Gruslys, Leader and Tan (2016).

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