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Frédéric Maffray

Publications and source records attributed to Frédéric Maffray.

At least 19 recordsLinked to original sources
arXiv2020

Coloring rings

A ring is a graph $R$ whose vertex set can be partitioned into $k \geq 4$ nonempty sets, $X_1, \dots, X_k$, such that for all $i \in \{1,\dots,k\}$, the set $X_i$ can be ordered as $X_i = \{u_i^1, \dots, u_i^{|X_i|}\}$ so that $X_i \subseteq N_R[u_i^{|X_i|}] \subseteq \dots \subseteq N_R[u_i^1] = X_{i-1} \cup X_i \cup X_{i+1}$. A hyperhole is a ring $R$ such that for all $i \in \{1,\dots,k\}$, $X_i$ is complete to $X_{i-1}\cup X_{i+1}$. In this paper, we prove that the chromatic number of a ring $R$ is equal to the maximum chromatic number of a hyperhole in $R$. Using this result, we give a polynomial-time coloring algorithm for rings. Rings formed one of the basic classes in a decomposition theorem for a class of graphs studied by Boncompagni, Penev, and Vušković in [Journal of Graph Theory 91 (2019), 192--246]. Using our coloring algorithm for rings, we show that graphs in this larger class can also be colored in polynomial time. Furthermore, we find the optimal $χ$-bounding function for this larger class of graphs, and we also verify Hadwiger's conjecture for it.

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arXiv2018

Polynomial Cases for the Vertex Coloring Problem

The computational complexity of the Vertex Coloring problem is known for all hereditary classes of graphs defined by forbidding two connected five-vertex induced subgraphs, except for seven cases. We prove the polynomial-time solvability of four of these problems: for ($P_5$, dart)-free graphs, ($P_5$, banner)-free graphs, ($P_5$, bull)-free graphs, and (fork, bull)-free graphs.

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arXiv2017

Coloring ($P_5$, bull)-free graphs

We give a polynomial-time algorithm that computes the chromatic number of any graph that contains no path on five vertices and no bull as an induced subgraph (where the bull is the graph with five vertices $a,b,c,d,e$ and edges $ab,bc,cd,be,ce$).

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arXiv2017

Decomposition techniques applied to the Clique-Stable set Separation problem

In a graph, a Clique-Stable Set separator (CS-separator) is a family $\mathcal{C}$ of cuts (bipartitions of the vertex set) such that for every clique $K$ and every stable set $S$ with $K \cap S = \emptyset$, there exists a cut $( W,W')$ in $\mathcal{C}$ such that $K \subseteq W$ and $S \subseteq W'$. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [17] asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive and negative results related to this question were given recently. Here we show how graph decomposition can be used to prove that a class of graphs admits a polynomial CS-separator. We apply this method to apple-free graphs and cap-free graphs.

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arXiv2017

Maximum Weight Stable Set in ($P_7$, bull)-free graphs and ($S_{1,2,3}$, bull)-free graphs

We give a polynomial time algorithm that finds the maximum weight stable set in a graph that does not contain an induced path on seven vertices or a bull (the graph with vertices $a$, $b$, $c$, $d$, $e$ and edges $ab$, $bc$, $cd$, $be$, $ce$). With the same arguments with also give a polynomial algorithm for any graph that does not contain $S_{1,2,3}$ or a bull.

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arXiv2017

Small feedback vertex sets in planar digraphs

Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.

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arXiv2016

Long induced paths in graphs

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $Ω(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on $n$ vertices, also contains an induced path on $Ω(\sqrt{\log n})$ vertices. We conjecture that for any $k$, there is a contant $c(k)$ such that any $k$-degenerate graph with a path on $n$ vertices also contains an induced path on $Ω((\log n)^{c(k)})$ vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs.

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arXiv2016

$χ$-bounded families of oriented graphs

A famous conjecture of Gyárfás and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star $S$ and integer $k$, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size $k$ or it contains $S$ as an induced subgraph. As an evidence, we prove that for any oriented star $S$, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order $3$ or $S$ as an induced subdigraph. We then study for which sets ${\cal P}$ of orientations of $P_4$ (the path on four vertices) similar statements hold. We establish some positive and negative results.

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arXiv2016

4-coloring ($P_6$, bull)-free graphs

We present a polynomial-time algorithm that determines whether a graph that contains no induced path on six vertices and no bull (the graph with vertices a, b, c, d, e and edges ab, bc, cd, be, ce) is 4-colorable. We also show that for any fixed k the k-coloring problem can be solved in polynomial time in the class of ($P_6$, bull, gem)-free graphs.

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arXiv2015

Graphs with no induced five-vertex path or antipath

We prove that a graph $G$ contains no induced $5$-vertex path and no induced complement of a $5$-vertex path if and only if $G$ is obtained from $5$-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement, where split unification is a new class-preserving operation introduced here.

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arXiv2015

On the choosability of claw-free perfect graphs

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most $4$.

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arXiv2015

On the b-chromatic number of the Cartesian product of two complete graphs

A b-coloring of a graph $G$ is a coloring of its vertices such that every color class contains a vertex that has neighbors in all other classes. The b-chromatic number of $G$ is the largest integer $k$ such that $G$ has a b-coloring with $k$ colors. Javadi and Omoomi ("On b-coloring of cartesian product of graphs", Ars Combinatoria 107 (2012) 521-536) proved that the b-chromatic number of $K_n \times K_n$ (the Cartesian product of two complete graphs on $n$ vertices) is in the set $\{2n-3, 2n-2\}$ and conjectured that the exact value is $2n-3$ for all $n \ge 5$. We give counterexamples to this conjecture for $n=5$, $n=6$ and $n=7$.

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arXiv2015

Equitable partition of graphs into induced forests

An equitable partition of a graph $G$ is a partition of the vertex-set of $G$ such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most $k$ colors can be equitably partitioned into $k-1$ induced forests. We also prove that for any integers $d\ge 1$ and $k\ge 3^{d-1}$, any $d$-degenerate graph can be equitably partitioned into $k$ induced forests. Each of these results implies the existence of a constant $c$ such that for any $k \ge c$, any planar graph has an equitable partition into $k$ induced forests. This was conjectured by Wu, Zhang, and Li in 2013.

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arXiv2015

Graphs with no induced wheel or antiwheel

A wheel is a graph that consists of a chordless cycle of length at least 4 plus a vertex with at least three neighbors on the cycle. It was shown recently that detecting induced wheels is an NP-complete problem. In contrast, it is shown here that graphs that contain no wheel and no antiwheel have a very simple structure and consequently can be recognized in polynomial time.

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arXiv2015

Even pairs in square-free Berge graphs with no odd prism

We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices). We prove that every graph G in this class either is a clique or has an even pair, as conjectured by Everett and Reed. This result is used to devise a polynomial-time algorithm to color optimally every graph in this class.

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arXiv2014

On color-critical ($P_{5},\overline{P}_5$)-free graphs

A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical $P_5$-free graphs for each $k \geq 5$. We show that the number of $k$-critical $(P_5, \overline{P}_5)$-free graphs is finite for every fixed $k$. Our result implies the existence of a certifying algorithm for $k$-coloring $(P_5, \overline{P}_5)$-free graphs.

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arXiv2013

On graphs with no induced subdivision of $K_4$

We prove a decomposition theorem for graphs that do not contain a subdivision of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four vertices. We obtain also a structure theorem for the class $\cal C$ of graphs that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in $\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\cal C$. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.

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