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Marthe Bonamy

Publications and source records attributed to Marthe Bonamy.

At least 73 records · Page 4Linked to original sources
arXiv2018

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 $k\geq Δ(G)$, while by Vizing's theorem, no more than $k=Δ(G)+1$ 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., $k=Δ(G)+2$ 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 $c:E\to\{1,2,\ldots,Δ(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$.

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arXiv2018

Bounding $χ$ by a fraction of $Δ$ for graphs without large cliques

The greedy coloring algorithm shows that a graph of maximum degree at most $Δ$ has chromatic number at most $Δ+ 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $Δ\geq 3$ and the graph contains no clique of size $Δ+ 1$. Reed's Conjecture states that the "greedy bound" can be improved by $k$ if the graph contains no clique of size $Δ+ 1 - 2k$. Johansson proved that the "greedy bound" can be improved by a factor of $Ω(\ln(Δ)^{-1})$ or $Ω\left(\frac{\ln(\ln(Δ))}{\ln(Δ)}\right)$ for graphs with no triangles or no cliques of any fixed size, respectively. Notably missing is a $\textit{linear}$ improvement on the "greedy bound" for graphs without large cliques. In this paper, we prove that for sufficiently large $Δ$, if $G$ is a graph with maximum degree at most $Δ$ and no clique of size $ω$, then $$χ(G) \leq 72Δ\sqrt{\frac{\ln(ω)}{\ln(Δ)}}.$$ This implies that for sufficiently large $Δ$, if $ω^{(72c)^2} \leq Δ$ then $χ(G) \leq Δ/c$. This bound actually holds for the list-chromatic and even the correspondence-chromatic number (also known as the DP-chromatic number). In fact, we prove what we call a "local version" of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. Our result simultaneously implies the linear improvement over the "greedy bound" and the two aforementioned results of Johansson.

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arXiv2017

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.

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arXiv2017

Independent Feedback Vertex Sets for Graphs of Bounded Diameter

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets $A$ and $B$, where $A$ is an independent set and $B$ induces a forest. The set $A$ in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

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arXiv2017

Independent Feedback Vertex Set for $P_5$-free Graphs

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $k\geq 0$, to delete at most $k$ vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the $3$-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for $H$-free graphs. We prove that it is NP-complete if $H$ contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $P_4$-free graphs. We show that it remains polynomial-time solvable for $P_5$-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most $k$ for a given integer $k\geq 0$. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for $H$-free graphs with the complexity of $3$-Colouring, Independent Odd Cycle Transversal and other related problems.

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arXiv2017

Tight lower bounds for the complexity of multicoloring

In the multicoloring problem, also known as ($a$:$b$)-coloring or $b$-fold coloring, we are given a graph G and a set of $a$ colors, and the task is to assign a subset of $b$ colors to each vertex of G so that adjacent vertices receive disjoint color subsets. This natural generalization of the classic coloring problem (the $b=1$ case) is equivalent to finding a homomorphism to the Kneser graph $KG_{a,b}$, and gives relaxations approaching the fractional chromatic number. We study the complexity of determining whether a graph has an ($a$:$b$)-coloring. Our main result is that this problem does not admit an algorithm with running time $f(b)\cdot 2^{o(\log b)\cdot n}$, for any computable $f(b)$, unless the Exponential Time Hypothesis (ETH) fails. A $(b+1)^n\cdot \text{poly}(n)$-time algorithm due to Nederlof [2008] shows that this is tight. A direct corollary of our result is that the graph homomorphism problem does not admit a $2^{O(n+h)}$ algorithm unless ETH fails, even if the target graph is required to be a Kneser graph. This refines the understanding given by the recent lower bound of Cygan et al. [SODA 2016]. The crucial ingredient in our hardness reduction is the usage of detecting matrices of Lindström [Canad. Math. Bull., 1965], which is a combinatorial tool that, to the best of our knowledge, has not yet been used for proving complexity lower bounds. As a side result, we prove that the running time of the algorithms of Abasi et al. [MFCS 2014] and of Gabizon et al. [ESA 2015] for the r-monomial detection problem are optimal under ETH.

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arXiv2016

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Let $G$ be a graph with a vertex colouring $α$. Let $a$ and $b$ be two colours. Then a connected component of the subgraph induced by those vertices coloured either $a$ or $b$ is known as a Kempe chain. A colouring of $G$ obtained from $α$ by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of $G$ are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. A conjecture of Mohar (2007) asserts that, for $k \geq 3$, all $k$-colourings of a $k$-regular graph that is not complete are Kempe equivalent. It was later shown that all $3$-colourings of a cubic graph that is neither $K_4$ nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each $k\geq 4$. We also report the implications of this result on the validity of the Wang-Swendsen-Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.

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arXiv2016

Token Sliding on Chordal Graphs

Let I be an independent set of a graph G. Imagine that a token is located on any vertex of I. We can now move the tokens of I along the edges of the graph as long as the set of tokens still defines an independent set of G. Given two independent sets I and J, the Token Sliding problem consists in deciding whether there exists a sequence of independent sets which transforms I into J so that every pair of consecutive independent sets of the sequence can be obtained via a token move. This problem is known to be PSPACE-complete even on planar graphs. In 2014, Demaine et al. asked whether the Token Sliding reconfiguration problem is polynomial time solvable on interval graphs and more generally in chordal graphs. Yamada and Uehara showed in 2016 that a polynomial time transformation can be found in proper interval graphs. In this paper, we answer the first question of Demaine et al. and generalize the result of Yamada and Uehara by showing that we can decide in polynomial time whether two independent sets of an interval graph are in the same connected component. Moveover, we answer similar questions by showing that: (i) determining if there exists a token sliding transformation between every pair of k-independent sets in an interval graph can be decided in polynomial time; (ii) deciding this problem becomes co-NP-hard and even co-W[2]-hard (parameterized by the size of the independent set) on split graphs, a sub-class of chordal graphs.

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arXiv2016

On the difference between the Szeged and Wiener index

We prove a conjecture of Nadjafi-Arani, Khodashenas and Ashrafi on the difference between the Szeged and Wiener index of a graph. Namely, if $G$ is a 2-connected non-complete graph on $n$ vertices, then $Sz(G)-W(G)\ge 2n-6$. Furthermore, the equality is obtained if and only if $G$ is the complete graph $K_{n-1}$ with an extra vertex attached to either $2$ or $n-2$ vertices of $K_{n-1}$. We apply our method to strengthen some known results on the difference between the Szeged and Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and Wiener index. We also propose a stronger version of the aforementioned conjecture.

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arXiv2015

List colouring with a bounded palette

Král' and Sgall (2005) introduced a refinement of list colouring where every colour list must be subset to one predetermined palette of colours. We call this $(k,\ell)$-choosability when the palette is of size at most $\ell$ and the lists must be of size at least $k$. They showed that, for any integer $k\ge 2$, there is an integer $C=C(k,2k-1)$, satisfying $C = O(16^{k}\ln k)$ as $k\to \infty$, such that, if a graph is $(k,2k-1)$-choosable, then it is $C$-choosable, and asked if $C$ is required to be exponential in $k$. We demonstrate it must satisfy $C = Ω(4^k/\sqrt{k})$. For an integer $\ell \ge 2k-1$, if $C(k,\ell)$ is the least integer such that a graph is $C(k,\ell)$-choosable if it is $(k,\ell)$-choosable, then we more generally supply a lower bound on $C(k,\ell)$, one that is super-polynomial in $k$ if $\ell = o(k^2/\ln k)$, by relation to an extremal set theoretic property. By the use of containers, we also give upper bounds on $C(k,\ell)$ that improve on earlier bounds if $\ell \ge 2.75 k$.

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arXiv2015

Linear kernels for outbranching problems in sparse digraphs

In the $k$-Leaf Out-Branching and $k$-Internal Out-Branching problems we are given a directed graph $D$ with a designated root $r$ and a nonnegative integer $k$. The question is to determine the existence of an outbranching rooted at $r$ that has at least $k$ leaves, or at least $k$ internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with $O(k^2)$ vertices are known on general graphs. In this work we show that $k$-Leaf Out-Branching admits a kernel with $O(k)$ vertices on $\mathcal{H}$-minor-free graphs, for any fixed family of graphs $\mathcal{H}$, whereas $k$-Internal Out-Branching admits a kernel with $O(k)$ vertices on any graph class of bounded expansion.

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arXiv2015

Incidence coloring of graphs with high maximum average degree

An incidence of an undirected graph G is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge of $G$ incident with $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent if one of the following holds: (i) $v = w$, (ii) $e = f$ or (iii) $vw = e$ or $f$. An incidence coloring of $G$ assigns a color to each incidence of $G$ in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama \emph{et al.}~\citep{ds05} proved that every graph with maximum average degree strictly less than $3$ can be incidence colored with $Δ+3$ colors. Recently, Bonamy \emph{et al.}~\citep{Bonamy} proved that every graph with maximum degree at least $4$ and with maximum average degree strictly less than $\frac{7}{3}$ admits an incidence $(Δ+1)$-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between $4$ and $6$. In particular we prove that every graph with maximum degree at least $7$ and with maximum average degree less than $4$ admits an incidence $(Δ+3)$-coloring. This result implies that every triangle-free planar graph with maximum degree at least $7$ is incidence $(Δ+3)$-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence $(Δ+ 7)$-coloring. More generally, we prove that $Δ+k-1$ colors are enough when the maximum average degree is less than $k$ and the maximum degree is sufficiently large.

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arXiv2014

A 13k-kernel for Planar Feedback Vertex Set via Region Decomposition

We show a kernel of at most $13k$ vertices for the Planar Feedback Vertex Set problem restricted to planar graphs, i.e., a polynomial-time algorithm that transforms an input instance $(G,k)$ to an equivalent instance with at most $13k$ vertices. To this end we introduce a few new reduction rules. However, our main contribution is an application of the region decomposition technique in the analysis of the kernel size. We show that our analysis is tight, up to a constant additive term.

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arXiv2014

Graphs with large chromatic number induce $3k$-cycles

Answering a question of Kalai and Meshulam, we prove that graphs without induced cycles of length $3k$ have bounded chromatic number. This implies the very first case of a much broader question asserting that every graph with large chromatic number induces a graph $H$ such that the sum of the Betti numbers of the independence complex of $H$ is also large.

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arXiv2014

The Erdős-Hajnal Conjecture for Long Holes and Anti-holes

Erdős and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$, there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size $n^{c_k}$.

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arXiv2014

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.

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arXiv2014

Planar graphs with $Δ\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

For 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 $Δ$-colorable or list total $(Δ+1)$-colorable, respectively, where $Δ$ is the maximum degree in the graph. It is already known that planar graphs with $Δ\geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $Δ\geq 7$ 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 $Δ\geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

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