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Marthe Bonamy

Publications and source records attributed to Marthe Bonamy.

At least 19 recordsLinked to original sources
arXiv2026

Meta-Theorems for Cuttable Distributed Problems

We prove that given any $\alpha$-approximation LOCAL algorithm for Minimum Dominating Set (MDS) on planar graphs, we can construct an $f(g)$-round $(3\alpha+1)$-approximation LOCAL algorithm for MDS on graphs embeddable in a given Euler genus-$g$ surface. Heydt et al. [European Journal of Combinatorics (2025)] gave an algorithm with $\alpha=11+\varepsilon$, from which we derive a $(34 +\varepsilon)$-approximation algorithm for graphs of genus $g$, therefore improving upon the current state of the art of $24g+O(1)$ due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of $91+\varepsilon$ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces. We generalize this result into two directions: (1) by considering other graph problems studied in Distributed Computing such as Minimum $k$-Tuple Dominating Set, for which constant-round approximation algorithms were known for planar graphs, but not for graphs of bounded genus; and (2) by considering graph classes beyond bounded genus graphs, called locally nice, and relying on the asymptotic dimension of the class. We prove these results by a series of meta-theorems about cuttable minimization problems with constant-round approximation LOCAL algorithms. Roughly speaking, in cuttable problems, one can systematically extract small subgraphs whose solutions are in proportion to the global solution restricted to the neighbourhood of the subgraph.

cs.DC↗
arXiv2026

A Gray code for arborescences of tournaments

We consider the following question of Knuth: given a directed graph $G$ and a root $r$, can the arborescences of $G$ rooted in $r$ be listed such that any two consecutive arborescences differ by only one arc? Such an ordering is called a pivot Gray code and can be formulated as a Hamiltonian path in the reconfiguration graph of the arborescences of $G$ under arc flips, also called flip graph of $G$. We give a positive answer for tournaments and explore several conditions showing that the flip graph of a directed graph may contain no Hamiltonian cycles.

math.CO↗
arXiv2025

Bipartite Tur\'an number of paths and other trees

We solve a recent question of Caro, Patk\'os and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of the bipartition. This was previously known only in the case where both sides of the bipartition have equal size and the longest path has size at most $5$. We also discuss possible generalizations replacing "path" with some specific types of trees.

math.CO↗
arXiv2025

On cuts of small chromatic number in sparse graphs

For a given integer $k$, let $\ell_k$ denote the supremum $\ell$ such that every sufficiently large graph $G$ with average degree less than $2\ell$ admits a separator $X \subseteq V(G)$ for which $\chi(G[X]) < k$. Motivated by the values of $\ell_1$, $\ell_2$ and $\ell_3$, a natural conjecture suggests that $\ell_k = k$ for all $k$. We prove that this conjecture fails dramatically: asymptotically, the trivial lower bound $\ell_k \geq \tfrac{k}{2}$ is tight. More precisely, we prove that for every $\varepsilon>0$ and all sufficiently large $k$, we have $\ell_k \leq (1+\varepsilon)\tfrac{k}{2}$.

math.CO↗
arXiv2025

Distributed Approximation Algorithms for Minimum Dominating Set in Locally Nice Graphs

We give a new, short proof that graphs embeddable in a given Euler genus-$g$ surface admit a simple $f(g)$-round $\alpha$-approximation distributed algorithm for Minimum Dominating Set (MDS), where the approximation ratio $\alpha \le 906$. Using tricks from Heydt et al. [European Journal of Combinatorics (2025)], we in fact derive that $\alpha \le 34 +\varepsilon$, therefore improving upon the current state of the art of $24g+O(1)$ due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of $91+\varepsilon$ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces. All our distributed algorithms work in the deterministic LOCAL model. They do not require any preliminary embedding of the graph and only rely on two things: a LOCAL algorithm for MDS on planar graphs with ``uniform'' approximation guarantees and the knowledge that graphs embeddable in bounded Euler genus surfaces have asymptotic dimension $2$. More generally, our algorithms work in any graph class of bounded asymptotic dimension where ``most vertices'' are locally in a graph class that admits a LOCAL algorithm for MDS with uniform approximation guarantees.

cs.DC↗
arXiv2025

On Modular Edge Colourings of Graphs

Given a graph $G$ and an integer $k\geq 2$, let $\chi'_k(G)$ denote the minimum number of colours required to colour the edges of $G$ such that, in each colour class, the subgraph induced by the edges of that colour has all non-zero degrees congruent to $1$ modulo $k$. In 1992, Pyber proved that $\chi'_2(G) \leq 4$ for every graph $G$, and posed the question of whether $\chi'_k(G)$ can be bounded solely in terms of $k$ for every $k\geq 3$. This question was answered in 1997 by Scott, who showed that $\chi'_k(G)\leq5k^2\log k$, and further asked whether $\chi'_k(G) = O(k)$. Recently, Botler, Colucci, and Kohayakawa (2023) answered Scott's question affirmatively proving that $\chi'_k(G) \leq 198k - 101$, and conjectured that the multiplicative constant could be reduced to $1$. A step towards this latter conjecture was made in 2024 by Nweit and Yang, who improved the bound to $\chi'_k(G) \leq 177k - 93$. In this paper, we further improve the multiplicative constant to $9$. More specifically, we prove that there is a function $f\in o(k)$ for which $\chi'_k(G) \leq 7k + f(k)$ if $k$ is odd, and $\chi'_k(G) \leq 9k + f(k)$ if $k$ is even. In doing so, we prove that $\chi'_k(G) \leq k + O(d)$ for every $d$-degenerate graph $G$, which plays a central role in our proof.

math.CO↗
arXiv2025

Infinite induced-saturated graphs

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$.

math.CO↗
arXiv2025

$\chi$-Boundedness and Neighbourhood Complexity of Bounded Merge-Width Graphs

Merge-width, recently introduced by Dreier and Toru\'nczyk, is a common generalisation of bounded expansion classes and twin-width for which the first-order model checking problem remains tractable. We prove that a number of basic properties shared by bounded expansion and bounded twin-width graphs also hold for bounded merge-width graphs: they are $\chi$-bounded, they satisfy the strong Erd\H{o}s-Hajnal property, and their neighbourhood complexity is linear.

math.CO↗
arXiv2025

Local Constant Approximation for Dominating Set on Graphs Excluding Large Minors

We show that graphs excluding $K_{2,t}$ as a minor admit a $f(t)$-round $50$-approximation deterministic distributed algorithm for Minimum Dominating Set. The result extends to Minimum Vertex Cover. Though fast and approximate distributed algorithms for such problems were already known for $H$-minor-free graphs, all of them have an approximation ratio depending on the size of $H$. To the best of our knowledge, this is the first example of a large non-trivial excluded minor leading to fast and constant-approximation distributed algorithms, where the ratio is independent of the size of $H$. A new key ingredient in the analysis of these distributed algorithms is the use of asymptotic dimension.

cs.DC↗
arXiv2025

On graph classes with constant domination-packing ratio

The dominating number $\gamma(G)$ of a graph $G$ is the minimum size of a vertex set whose closed neighborhood covers all the vertices of the graph. The packing number $\rho(G)$ of $G$ is the maximum size of a vertex set whose closed neighborhoods are pairwise disjoint. In this paper we study graph classes ${\cal G}$ such that $\gamma(G)/\rho(G)$ is bounded by a constant $c_{\cal G}$ for each $G\in {\cal G}$. We propose an inductive proof technique to prove that if $\cal G$ is the class of $2$-degenerate graphs, then there is such a constant bound $c_{\cal G}$. We note that this is the first monotone, dense graph class that is shown to have constant ratio. We also show that the classes of AT-free and unit-disk graphs have bounded ratio. In addition, our technique gives improved bounds on $c_{\cal G}$ for planar graphs, graphs of bounded treewidth or bounded twin-width. Finally, we provide some new examples of graph classes where the ratio is unbounded.

math.CO↗
arXiv2024

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

A graph is $\mathcal{O}_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $\mathcal{O}_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of $\mathcal{O}_2$-free graphs without $K_{2,3}$ as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in $\mathcal{O}_k$-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $\mathcal{O}_k$-free graphs, and that deciding the $\mathcal{O}_k$-freeness of sparse graphs is polynomial time solvable.

math.CO↗
arXiv2023

Edge-colouring graphs with local list sizes

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph $G$ with sufficiently large maximum degree $Δ$ and minimum degree $δ\geq \ln^{25} Δ$, the following holds: for every assignment of lists of colours to the edges of $G$, such that $|L(e)| \geq (1+o(1)) \cdot \max\left\{\rm{deg}(u),\rm{deg}(v)\right\}$ for each edge $e=uv$, there is an $L$-edge-colouring of $G$. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, $k$-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.

math.CO↗
arXiv2023

Separating the edges of a graph by a linear number of paths

Recently, Letzter proved that any graph of order $n$ contains a collection $\mathcal{P}$ of $O(n\log^\star n)$ paths with the following property: for all distinct edges $e$ and $f$ there exists a path in $\mathcal{P}$ which contains $e$ but not $f$. We improve this upper bound to $19 n$, thus answering a question of G.O.H. Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhár and by Falgas-Ravry, Kittipassorn, Korándi, Letzter, and Narayanan. Our proof is elementary and self-contained.

math.CO↗
arXiv2023

Partitioning edges of a planar graph into linear forests and a matching

We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree $Δ\ge 9$ can be partitioned into $\tfrac{Δ-1}{2}$ linear forests and one matching. This strengthens well-known results stating that graphs in this class have chromatic index $Δ$ [Vizing, 1965] and linear arboricity at most $\lceil(Δ+1)/2\rceil$ [Wu, 1999].

math.CO↗
arXiv2022

Every planar graph with $Δ\geqslant 8$ is totally $(Δ+2)$-choosable

Total coloring is a variant of edge coloring where both vertices and edges are to be colored. A graph is totally $k$-choosable if for any list assignment of $k$ colors to each vertex and each edge, we can extract a proper total coloring. In this setting, a graph of maximum degree $Δ$ needs at least $Δ+1$ colors. In the planar case, Borodin proved in 1989 that $Δ+2$ colors suffice when $Δ$ is at least 9. We show that this bound also holds when $Δ$ is $8$.

cs.DM↗
arXiv2022

A tight local algorithm for the minimum dominating set problem in outerplanar graphs

We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a $(5-\varepsilon)$-approximation, for any $\varepsilon>0$. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.

cs.DC↗
arXiv2022

Tuza's Conjecture for Threshold Graphs

Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.

math.CO↗