Marthe Bonamy
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Marthe Bonamy has a total of 10 co-authors. Here are the top co-authors by number of shared publications:
- Alexandre Pinlou
Université de Montpellier CNRS LIRMM
3 shared publications - Benjamin Lévêque
Département MIAp LIRMM Université Montpellier 2 CNRS Université Paul-Valéry Montpellier 3
3 shared publications - Nicolas Bousquet
EDF R&D Chatou France
3 shared publications - Daniël Paulusma
Durham University, Durham, UK
2 shared publications - Ioannis Lignos
Durham University
2 shared publications - Matthew Johnson
Durham University
2 shared publications
Latest Publications
- Open Access
On the neighbour sum distinguishing index of planar graphs
Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=\Delta(G)+2$ enables enforcing additional strong feature of $c$, namely that it attributes distinct sums of incident colours to adjacent vertices in $G$ if only this graph has no isolated edges and is not isomorphic to $C_5$. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph $G$ of maximum degree at least $28$ which contains no isolated edges admits a proper edge colouring $c:E\to\{1,2,\ldots,\Delta(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$.Open AccessOn the neighbour sum distinguishing index of planar graphs
Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating irregularities in graphs, it has been moreover conjectured that only slightly larger $k$, i.e., $k=\Delta(G)+2$ enables enforcing additional strong feature of $c$, namely that it attributes distinct sums of incident colours to adjacent vertices in $G$ if only this graph has no isolated edges and is not isomorphic to $C_5$. We prove the conjecture is valid for planar graphs of sufficiently large maximum degree. In fact even stronger statement holds, as the necessary number of colours stemming from the result of Vizing is proved to be sufficient for this family of graphs. Specifically, our main result states that every planar graph $G$ of maximum degree at least $28$ which contains no isolated edges admits a proper edge colouring $c:E\to\{1,2,\ldots,\Delta(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$. - 18 December 2015
- Published by Wiley in Journal of Graph Theory
- Vol. 84 (1) , 93-103
List Coloring with a Bounded Palette
- 18 December 2015
- Published by Wiley in Journal of Graph Theory
- Vol. 84 (1) , 93-103
List Coloring with a Bounded Palette
- Open Access
Reconfiguring Independent Sets in Cographs
Two independent sets of a graph are adjacent if they differ on exactly one vertex (i.e. we can transform one into the other by adding or deleting a vertex). Let $k$ be an integer. We consider the reconfiguration graph $TAR_k(G)$ on the set of independent sets of size at least $k$ in a graph $G$, with the above notion of adjacency. Here we provide a cubic-time algorithm to decide whether $TAR_k(G)$ is connected when $G$ is a cograph, thus solving an open question of~[Bonsma 2014]. As a by-product, we also describe a linear-time algorithm which decides whether two elements of $TAR_k(G)$ are in the same connected component.Open AccessReconfiguring Independent Sets in Cographs
Two independent sets of a graph are adjacent if they differ on exactly one vertex (i.e. we can transform one into the other by adding or deleting a vertex). Let $k$ be an integer. We consider the reconfiguration graph $TAR_k(G)$ on the set of independent sets of size at least $k$ in a graph $G$, with the above notion of adjacency. Here we provide a cubic-time algorithm to decide whether $TAR_k(G)$ is connected when $G$ is a cograph, thus solving an open question of~[Bonsma 2014]. As a by-product, we also describe a linear-time algorithm which decides whether two elements of $TAR_k(G)$ are in the same connected component. - Open AccessPlanar graphs with $Δ\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosableFor planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list edge $\Delta$-colorable or list total $(\Delta+1)$-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta\geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta\geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k$ ($\forall 3 \leq k \leq 6$) are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.Open AccessPlanar graphs with $Δ\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosableFor planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list edge $\Delta$-colorable or list total $(\Delta+1)$-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta\geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta\geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k$ ($\forall 3 \leq k \leq 6$) are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.
- Open Access
Recoloring graphs via tree decompositions
Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent proper $k$-colorings. Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum degree plus two. We first improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring. Any graph is $(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012). We also prove that given any two $(\chi(G)+1)$-colorings of a cograph (resp. distance-hereditary graph) $G$, we can find a linear (resp. quadratic) sequence between them. In both cases, the bounds cannot be improved by more than a constant factor for a fixed $\chi(G)$. The graph classes are also optimal in some sense: one of the smallest interesting superclass of distance-hereditary graphs corresponds to comparability graphs, for which no such property holds (even when relaxing the constraint on the length of the sequence). As for cographs, they are equivalently the graphs with no induced $P_4$, and there exist $P_5$-free graphs that admit no sequence between two of their $(\chi(G)+1)$-colorings. All the proofs are constructivist and lead to polynomial-time recoloring algorithmOpen AccessRecoloring graphs via tree decompositions
Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent proper $k$-colorings. Jerrum proved that any graph is $k$-mixing if $k$ is at least the maximum degree plus two. We first improve Jerrum's bound using the grundy number, which is the worst number of colors in a greedy coloring. Any graph is $(tw+2)$-mixing, where $tw$ is the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two $(tw+2)$-colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012). We also prove that given any two $(\chi(G)+1)$-colorings of a cograph (resp. distance-hereditary graph) $G$, we can find a linear (resp. quadratic) sequence between them. In both cases, the bounds cannot be improved by more than a constant factor for a fixed $\chi(G)$. The graph classes are also optimal in some sense: one of the smallest interesting superclass of distance-hereditary graphs corresponds to comparability graphs, for which no such property holds (even when relaxing the constraint on the length of the sequence). As for cographs, they are equivalently the graphs with no induced $P_4$, and there exist $P_5$-free graphs that admit no sequence between two of their $(\chi(G)+1)$-colorings. All the proofs are constructivist and lead to polynomial-time recoloring algorithm - 27 February 2014
- Published by Wiley in Journal of Graph Theory
- Vol. 77 (3) , 190-218
2‐Distance Coloring of Sparse Graphs
For graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2‐distance ‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance ‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than and with large enough maximum degree Δ (depending only on m) can be list 2‐distance ‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance ‐colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2‐distance ‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less.
- 27 February 2014
- Published by Wiley in Journal of Graph Theory
- Vol. 77 (3) , 190-218
2‐Distance Coloring of Sparse Graphs
For graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2‐distance ‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance ‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than and with large enough maximum degree Δ (depending only on m) can be list 2‐distance ‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance ‐colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2‐distance ‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less.
- Open Access
Brooks' theorem on powers of graphs
We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\Delta \geq 3$ can be lowered by 1, even in the case of online list coloring.Open AccessBrooks' theorem on powers of graphs
We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\Delta \geq 3$ can be lowered by 1, even in the case of online list coloring. - Open Access
List coloring the square of sparse graphs with large degree
We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree $\Delta$ are list $(\Delta+1)$-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs of girth at least $6$ are of maximum average degree less than $3$, and planar graphs of girth at least $7$ are of maximum average degree less than $14/5<3$. We strengthen their result and prove that there exists a function $f$ such that the square of any graph with maximum average degree $m<3$ and maximum degree $\Delta\geq f(m)$ is list $(\Delta+1)$-colorable. This bound of $3$ is optimal in the sense that the above-mentioned planar graphs with girth $6$ have maximum average degree less than $3$ and arbitrarily large maximum degree, while their square cannot be $(\Delta+1)$-colored. The same holds for list injective $\Delta$-coloring.Open AccessList coloring the square of sparse graphs with large degree
We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree $\Delta$ are list $(\Delta+1)$-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs of girth at least $6$ are of maximum average degree less than $3$, and planar graphs of girth at least $7$ are of maximum average degree less than $14/5<3$. We strengthen their result and prove that there exists a function $f$ such that the square of any graph with maximum average degree $m<3$ and maximum degree $\Delta\geq f(m)$ is list $(\Delta+1)$-colorable. This bound of $3$ is optimal in the sense that the above-mentioned planar graphs with girth $6$ have maximum average degree less than $3$ and arbitrarily large maximum degree, while their square cannot be $(\Delta+1)$-colored. The same holds for list injective $\Delta$-coloring. - Open Access
Planar graphs with maximum degree D at least 8 are (D+1)-edge-choosable
We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every graph is (D+1)-edge-choosable, where D is the maximum degree. In 1990, Borodin solved the conjecture for planar graphs with maximum degree at least 9, and asked whether the bound could be lowered to 8. We prove here that planar graphs with maximum degree D at least 8 are (D+1)-edge-choosable.
Open AccessPlanar graphs with maximum degree D at least 8 are (D+1)-edge-choosable
We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every graph is (D+1)-edge-choosable, where D is the maximum degree. In 1990, Borodin solved the conjecture for planar graphs with maximum degree at least 9, and asked whether the bound could be lowered to 8. We prove here that planar graphs with maximum degree D at least 8 are (D+1)-edge-choosable.
- 26 April 2012
- Published by Springer Nature in Journal of Combinatorial Optimization
- Vol. 27 (1) , 132-143
Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs
- 26 April 2012
- Published by Springer Nature in Journal of Combinatorial Optimization
- Vol. 27 (1) , 132-143
Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs