Julien Bensmail

• Université Côte d'Azur

In a recent work, Keusch proved the so-called 1-2-3 Conjecture, raised by Karo\'nski, {\L}uczak, and Thomason in 2004: for every connected graph different from $K_2$, we can assign labels~$1,2,3$ to the edges so that no two adjacent vertices are incident to the same sum of labels. Despite this significant result, several problems close to the 1-2-3...

Locally irregular graphs are those graphs in which any two adjacent vertices have different degrees, while locally irregular decompositions are edge-partitions of graphs where each part induces a locally irregular graph. These notions were introduced in a seminal work of Baudon et al., in connection, in particular, to the so-called 1-2-3 Conjecture...

We deal with a variant of the 1–2–3 Conjecture introduced by Gao, Wang, and Wu (Graphs Combin 32:1415–1421, 2016) . This variant asks whether all graphs can have their edges labelled with 1 and 2 so that when computing the sums of labels incident to the vertices, no monochromatic cycle appears. In the aforementioned seminal work, the authors mainly...

A well-known result of Bondy and Chvátal establishes that a graph of order \(n\) is Hamiltonian if and only if its \(n\)-closure (obtained through repeatedly adding an edge joining any two non-adjacent vertices with degree sum at least \(n\)) also is. In this work, we investigate such closure results for arbitrarily partitionable graphs, a weakenin...

A (di)graph H has the Erdős-Pósa (EP) property for (butterfly) minors if there exists a function \(f: \mathbb {N} \rightarrow \mathbb {N}\) such that, for any \(k\in \mathbb {N}\) and any (di)graph G, either G contains at least k pairwise vertex-disjoint copies of H as (butterfly) minor, or there exists a subset T of at most f(k) vertices such that...

We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kaziów in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1, 2, 3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.

The (directed) metric dimension of a digraph D, denoted by \({{\,\mathrm{\overrightarrow{\textrm{MD}}}\,}}(D)\), is the size of a smallest subset S of vertices such that every two vertices of D are distinguished via their distances from the vertices in S. In this paper, we investigate the graph parameters \({{\,\textrm{BOMD}\,}}(G)\) and \({{\,\tex...

The Weak (2, 2)-Conjecture is a graph labelling problem asking whether all connected graphs of at least three vertices can have their edges assigned red labels 1 and 2 and blue labels 1 and 2 so that any two adjacent vertices are distinguished either by their sums of incident red labels, or by their sums of incident blue labels. This problem emerge...

In the so-called 1-2-3 Conjecture, the question is, for any connected graph not isomorphic to K2, whether we can label its edges with 1,2,3 so that no two adjacent vertices are incident to the same sum of labels. Many aspects of this conjecture have been investigated over the last past years, related both to the conjecture itself and to variations...

Let G be a graph, and ℓ:E(G)→{1,…,k} be a k-labelling of G, i.e., an assignment of labels from {1,…,k} to the edges of G. We say that ℓ is irregular if no two distinct vertices of G are incident to the same sum of labels. The irregularity strength of G, denoted by s(G), is the smallest k such that irregular k-labellings of G exist. These notions we...

A proper k-edge-colouring ϕ of a graph G is an assignment of colours from {1,…,k} to the edges of G such that no two adjacent edges receive the same colour. If, additionally, ϕ guarantees that no two adjacent vertices of G are incident to the same sets or sums of colours, then ϕ is called an AVD or NSD edge-colouring, respectively (the abbreviation...

This paper introduces the largest connected subgraph game played on an undirected graph G. In each round, Alice first colours an uncoloured vertex of G red, and then, Bob colours an uncoloured vertex of G blue, with all vertices initially uncoloured. Once all the vertices are coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connect...

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with 1, 2, 3 so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 C...

The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been investigated in the literature, including variants for modifications of proper colourings. These variants include, nota...

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture, that may be attributed to Chartrand et al., is that almost ever...

In this paper, we study the recently introduced scoring game played on graphs called the Edge-Balanced Index Game. This game is played on a graph by two players, Alice and Bob, who take turns colouring an uncoloured edge of the graph. Alice plays first and colours edges red, while Bob colours edges blue. The game ends once all the edges have been c...

A signed graph (G,σ) is a graph G along with a function σ:E(G)→{+,−}. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks....

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support...

We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kaziów in 2012, is true. Namely, for every connected graph with order at least 3, we can assign labels 1, 2, 3 to the edges so that no two adjacent vertices are incident to the same product of labels.

The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been investigated in the literature, including variants for modifications of proper colourings. These variants include, nota...

We introduce the largest connected subgraph game played on an undirected graph G. Each round, Alice colours an uncoloured vertex of G red, and then, Bob colours one blue. Once every vertex is coloured, Alice (Bob, resp.) wins if there is a red (blue, resp.) connected subgraph whose order is greater than that of any blue (red, resp.) connected subgr...

The 1-2-3 Conjecture states that every connected graph different from K2 admits a proper 3-(edge-)labelling, i.e., can have its edges labelled with 1,2,3 so that no two adjacent vertices are incident to the same sum of labels. In connection with some recent optimisation variants of this conjecture, in this paper we investigate the role of the label...

We prove that the product version of the 1-2-3 Conjecture, raised by Skowronek-Kazi{\'o}w in 2012, is true. Namely, for every connected graph with order at least 3, we prove that we can assign labels 1,2,3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.

In this work, we consider equitable proper labellings of graphs, which were recently introduced by Baudon, Pilśniak, Przybyło, Senhaji, Sopena, and Woźniak. Given a graph G, the goal is to assign labels to the edges so that (1) no two adjacent vertices are incident to the same sum of labels, and (2) every two labels are assigned about the same numb...

The 1-2-3 Conjecture states that every connected graph different from \(K_2\) admits a proper 3-labelling, i.e., can have its edges labelled with 1, 2, 3 so that no two adjacent vertices are incident to the same sum of labels. In connection with recent optimisation variants of this conjecture, we study the role of label 3 in proper 3-labellings of...

Given an undirected graph, in the AVD (edge-colouring) Conjecture, the goal is to find a proper edge-colouring with the least number of colours such that every two adjacent vertices are incident to different sets of colours. More precisely, the conjecture says that, a few exceptions apart, every graph G should admit such an edge-colouring with at m...

International audience In a recent work, Bensmail, Blanc, Cohen, Havet and Rocha, motivated by applications for TDMA scheduling problems, have introduced the notion of BMRN*-colouring of digraphs, which is a type of arc-colouring with particular colouring constraints. In particular, they gave a special focus to planar digraphs. They notably proved...

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with $1,2,3$ so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 C...

In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get in...

Pushable homomorphisms and the pushable chromatic number χp of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G⃗, we have χp(G⃗)≤χo(G⃗)≤2χp(G⃗), where χo(G⃗) denotes the oriented chromatic number of G⃗. This stands as the first general bounds on χp. This parameter was fur...

A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition \(E_1,\dots ,E_k\) of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conje...

In the localization game, introduced by Seager in 2013, an invisible and immobile target is hidden at some vertex of a graph G. At every step, one vertex v of G can be probed which results in the knowledge of the distance between v and the secret location of the target. The objective of the game is to minimize the number of steps needed to locate t...

A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies an...

The metric dimension MD(G) of an undirected graph G is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of G. Many aspects of this notion have been investigated since its introduction in the 70’s, including its generalization to digraphs. In this work, we study, for...

The 1-2-3 Conjecture states that every nice graph G (without component isomorphic to \(K_2\)) admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture is that every nice graph G can be turned into a locally irregular...

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong sup...

Horňak, Przybyło and Woźniak recently proved that, a small class of obvious exceptions apart, every digraph can be 4-arc-weighted so that, for every arc uv⃗, the sum of weights incoming to u is different from the sum of weights outgoing from v. They conjectured a stronger result, namely that the same statement with 3 instead of 4 should also be tru...

A graph G of order n is arbitrarily partitionable (AP) if, for every sequence (n1, . . ., np) partitioning n, there is a partition (V1, . . ., ,Vp) of V (G) such that G[Vi] is a connected ni-graph for i = 1, . . ., p. The property of being AP is related to other well-known graph notions, such as perfect matchings and Hamiltonian cycles, with which...

The 1–2–3 Conjecture asserts that, for every connected graph different from K2, its edges can be labeled with 1,2,3 so that, when coloring each vertex with the sum of its incident labels, no two adjacent vertices get the same color. This conjecture takes place in the more general context of distinguishing labelings, where the goal is to label graph...

A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the $80$'s, asserts that every graph with m...

Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{...

Neighbour-sum-distinguishing edge-weightings are a way to "encode" proper vertex-colourings via the sums of weights incident to the vertices. Over the last decades, this notion has been attracting, in the context of several conjectures, ingrowing attention dedicated, notably, to understanding, which weights are needed to produce neighbour-sum-disti...

The metric dimension MD(G) of an undirected graph G is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of G. Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for...

We investigate graph colouring models for the purpose of optimizing TDMA link scheduling in Wireless Networks. Inspired by the BPRN-colouring model recently introduced by Rocha and Sasaki, we introduce a new colouring model, namely the BMRN-colouring model, which can be used to model link scheduling problems where particular types of collisions mus...

How can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An...

For any S ⊂ ℤ we say that a graph G has the S-property if there exists an S-edge-weighting w : E(G) → S such that for any pair of adjacent vertices u, v we have ∑e∈E(v) w(e) ≠ ∑e∈E(u) w(e), where E(v) and E(u) are the sets of edges incident to v and u, respectively. This work focuses on {a, a + 2}-edge-weightings where a ∈ ℤ is odd. We show that a...

The 1-2-3 Conjecture, posed by Karoński, Łuczak and Thomason, asks whether every connected graph [Formula presented] different from [Formula presented] can be 3-edge-weighted so that every two adjacent vertices of [Formula presented] get distinct sums of incident weights. Towards that conjecture, the best-known result to date is due to Kalkowski, K...

An error is spotted in the statement of Theorem 1.3 of our published article titled “On oriented cliques with respect to push operation” (Discrete Applied Mathematics 2017). The theorem provided an exhaustive list of 16 minimal (up to spanning subgraph inclusion) underlying planar push cliques. The error was that, one of the 16 graphs from the abov...

An error is spotted in the statement of Theorem~1.3 of our published article titled "On oriented cliques with respect to push operation" (Discrete Applied Mathematics 2017). The theorem provided an exhaustive list of 16 minimal (up to spanning subgraph inclusion) underlying planar push cliques. The error was that, one of the 16 graphs from the abov...

The 1-2 Conjecture raised by Przybyło and Woźniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights “incident” to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally ir...

Seager introduced the following game in 2013. An invisible and immobile target is hidden at some vertex of a graph G. Every step, one vertex v of G can be probed which results in the knowledge of the distance between v and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is. We addres...

Due to a classical result of Berge, it is known that a matching of any graph can be turned into a maximum matching by repeatedly augmenting alternating paths whose ends are not covered. In a recent work, Nisse, Salch and Weber considered the influence, on this process, of augmenting paths with length at most k only. Given a graph G, an initial matc...

A graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E1,...,Ek of E(G) such that each G[Ei] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail...

The 1–2–3 Conjecture asks whether every graph with no connected component isomorphic to K2 can be 3-edge-weighted so that every two adjacent vertices u and v can be distinguished via the sum of their incident weights, that is the incident sums of u and v differ by at least 1. We here investigate the consequences on the 1–2–3 Conjecture of requiring...

A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if the edges can be extended to an arrangement of pseudolines. In the recent study of crossing numbers, pseudolin...

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every $d$-regular graph, $d\geq 2$, can be decomposed into at most $2$ subgraphs (without isolated edges) fulfilling th...

The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every $d$-regular graph, $d\geq 2$, can be decomposed into at most $2$ subgraphs (without isolated edges) fulfilling th...

How can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An...

The 1-2-3 Conjecture asks whether every graph with no connected component isomorphic to $K_2$ can be $3$-edge-weighted so that every two adjacent vertices $u$ and $v$ can be distinguished via the sum of their incident weights, that is the incident sums of $u$ and $v$ differ by at least~$1$. We here investigate the consequences on the 1-2-3 Conjec...

A (m,n)-colored mixed graph is a graph having arcs of m different colors and edges of n different colors. A graph homomorphism of a (m,n)-colored mixed graph G to a (m,n)-colored mixed graph H is a vertex mapping such that if uv is an arc (edge) of color c in G, then f(u)f(v) is also an arc (edge) of color c. The (m,n)-colored mixed chromatic numbe...

The dichromatic number χ(D) of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chr...

The 1-2-3 Conjecture asks whether every graph with no connected component isomorphic to K_2 can be 3-edge-weighted so that every two adjacent vertices u and v can be distinguished via the sum of their incident weights, that is the incident sums of u and v differ by at least 1. We here investigate the consequences on the 1-2-3 Conjecture of requirin...

This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights $1,...,|E(G)|$ to the edges of any given graph $G$ (with no component isomorphic to $K_2$) so that every two adjacent vertices of $G$ get distinguished by their sums of incident weights? One may see this question as a combination of t...

This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights $1,...,|E(G)|$ to the edges of any given graph $G$ (with no component isomorphic to $K_2$) so that every two adjacent vertices of $G$ get distinguished by their sums of incident weights? One may see this question as a combination of t...

In the context of a conjecture of Erdos and Gy'arf'as, we consider, for any q = 2, the existence of q-power cycles (i.e., with length a power of q) in cubic graphs. We exhibit constructions showing that, for every q = 3, there exist arbitrarily large cubic graphs with no q-power cycles. Concerning the remaining case q = 2 (which corresponds to the...

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition E_1 \cup E_2 \cup ... \cup E_k of the edge set E(G) such that each E_i induces a locally irregular graph. It was recently conjectured by Baudon et al. that...

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all graphs have such a decomposition, but Baudon, Bensmail, Przyby{\l}o, and Wo\'zniak conjectured that if $G$ can be...

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all graphs have such a decomposition, but Baudon, Bensmail, Przyby{\l}o, and Wo\'zniak conjectured that if $G$ can be...

The Bar\'at-Thomassen conjecture asserts that for every tree $T$ on $m$ edges, there exists a constant $k_T$ such that every $k_T$-edge-connected graph with size divisible by $m$ can be edge-decomposed into copies of $T$. So far this conjecture has only been verified when $T$ is a path or when $T$ has diameter at most 4. Here we prove the full stat...

The Bar\'at-Thomassen conjecture asserts that for every tree $T$ on $m$ edges, there exists a constant $k_T$ such that every $k_T$-edge-connected graph with size divisible by $m$ can be edge-decomposed into copies of $T$. So far this conjecture has only been verified when $T$ is a path or when $T$ has diameter at most 4. Here we prove the full stat...

International audience A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graph $G$, there exists a minimum integer $\chi^{\prime}_{\mathrm{irr}}(G)$ such that $G$ admits an edge-par...

International audience An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\o...

A strong edge-coloring of a graph G is an assignment of colors to edges such that every color class induces a matching. We here focus on bipartite graphs whose one part is of maximum degree at most 3 and the other part is of maximum degree Δ. For every such graph, we prove that a strong 4Δ-edge-coloring can always be obtained. Together with a resul...

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that r...

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that r...

A strong edge coloring of a graph is a proper edge coloring such that no edge has two incident edges of the same color. Erdős and Nešetřil conjectured in 1989 that colors are always enough for a strong edge coloring, where Δ is the maximum degree of the graph. In the specific case where , we prove this to be true when there is no subgraph with aver...

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph can be decomposed into locally irregular subgraphs if its edge set may be partitioned into subsets each of which induces a locally irregular subgraph in . We characterize all connected graphs which cannot be decomposed into locally irregular subg...

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \g...