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Nicolas Trotignon

Publications and source records attributed to Nicolas Trotignon.

83 records · Page 5Linked to original sources
arXiv2013

Algorithms for perfectly contractile graphs

We consider the class ${\cal A}$ of graphs that contain no odd hole, no antihole of length at least 5, and no "prism" (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class ${\cal A}'$ of graphs that contain no odd hole, no antihole of length at least 5 and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are relevant to the study of perfect graphs. We give polynomial-time recognition algorithms for these two classes. We proved previously that every graph $G\in{\cal A}$ is "perfectly contractile", as conjectured by Everett and Reed [see the chapter "Even pairs" in the book {\it Perfect Graphs}, J.L. Ram\'ırez-Alfons\'ın and B.A. Reed, eds., Wiley Interscience, 2001]. The analogous conjecture concerning graphs in ${\cal A}'$ is still open.

cs.DM↗
arXiv2013

Graphes parfaits : structure et algorithmes

This work is the PhD thesis of Nicolas Trotignon, written in 2004 under the supervision of Frédéric Maffray. It is motivated by the desire for a better understanding of perfect graphs. The proof of the Claude Berge's perfect graph conjecture in 2002 by Chudnovsky, Robertson, Seymour and Thomas has shed a new light on this field of combinatorics. But some questions are still unsettled, particulary the existence of a combinatorial algorithm for the coloring of perfect graphs. An even pair of a graph is a pair of vertices such that every path joining them has even length. As proved by Fonlupt and Uhry, the contraction of an even pair preserves the chromatic number, and when applied recursively may lead to an optimal coloring. We prove a conjecture of Everett and Reed saying that this method works for a class of perfect graphs: Artemis graphs. This yields a coloring algorithm for Artemis graphs with complexity $O(n^2m)$. We give an $O(n^9)$ algorithm for the recognition of Artemis graphs. Other recognition algorithms are also given, each of them based on subgraph detection routines for Berge graphs. We show that these subgraph detection problems are NP-complete when extended to general graphs.

math.CO↗
arXiv2013

Detecting wheels

A \emph{wheel} is a graph made of a cycle of length at least~4 together with a vertex that has at least three neighbors in the cycle. We prove that the problem whose instance is a graph $G$ and whose question is "does $G$ contains a wheel as an induced subgraph" is NP-complete. We also settle the complexity of several similar problems.

cs.DM↗
arXiv2013

On the structure of self-complementary graphs

A \emph{self-complementary} graph is a graph isomorphic to its complement. An isomorphism between $G$ and its complement, viewed as a permutation of $V(G)$, is then called an \emph{antimorphism}. A \emph{skew partition} of $G$ is a partition of $V(G)$ into 4 sets $A,B,C,D$ such that there is no edge between $A,B$ and every possible edge between $C,D$. A \emph{symmetric partition} of $G$ is a partition of $V(G)$ into 4 sets $A,B,C,D$ such that there is no edge between $A, D$, no edge between $B, C$, every possible edge between $A,B$ and every possible edge between $C,D$. We give a new proof of a theorem of Gibbs saying that every self-complementary graph on $4k$ vertices has $k$ disjoint paths on 4 vertices as induced subgraph. This new proof gives more structural information than the original one. We conjecture that every self-complementary graph on $4k$ vertices either has an induced cycle on 5 vertices, or a skew partition, or a symmetric partition. The new proof of Gibb's theorem yields a proof of the conjecture for the self-complementary graphs that have an antimorphism that is the product of a two circular permutations, one of them of length 4.

math.CO↗
arXiv2013

Ramsey-type results on singletons, co-singletons and monotone sequences in large collections of sets

We say that a 0-1 matrix $N$ of size $a\times b$ can be found in a collection of sets $\mathcal{H}$ if we can find sets $H_{1}, H_{2}, \dots, H_{a}$ in $\mathcal{H}$ and elements $e_1, e_2, \dots, e_b$ in $\cup_{H \in \mathcal{H}} H$ such that $N$ is the incidence matrix of the sets $H_{1}, H_{2}, \dots, H_{a}$ over the elements $e_1, e_2, \dots, e_b$. We prove the following Ramsey-type result: for every $n\in \N$, there exists a number S(n) such that in any collection of at least S(n) sets, one can find either the incidence matrix of a collection of $n$ singletons, or its complementary matrix, or the incidence matrix of a collection of $n$ sets completely ordered by inclusion. We give several results of the same extremal set theoretical flavour. For some of these, we give the exact value of the number of sets required.

math.CO↗
arXiv2013

Complements of nearly perfect graphs

A class of graphs closed under taking induced subgraphs is $χ$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $χ(G) \leq f(ω(G))$. We consider the following question initially studied in [A. Gy{á}rf{á}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $χ$-bounded class $\cal C$, is the class $\bar{C}$ $χ$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $χ$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $χ$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $χ$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $χ$-bounded. For every $j$, we construct a class of graphs $χ$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $χ$-bounded.

math.CO↗
arXiv2013

Substitution and $χ$-Boundedness

A class $\mathcal{G}$ of graphs is said to be {\em $χ$-bounded} if there is a function $f:\mathbb{N} \rightarrow \mathbb{R}$ such that for all $G \in \mathcal{G}$ and all induced subgraphs $H$ of $G$, $χ(H) \leq f(ω(H))$. In this paper, we show that if $\mathcal{G}$ is a $χ$-bounded class, then so is the closure of $\mathcal{G}$ under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if $\mathcal{G}$ is $χ$-bounded by a polynomial (respectively: exponential) function, then the closure of $\mathcal{G}$ under substitution is also $χ$-bounded by some polynomial (respectively: exponential) function. In addition, we show that if $\mathcal{G}$ is a $χ$-bounded class, then the closure of $\mathcal{G}$ under the operations of gluing along a clique and gluing along a bounded number of vertices together is also $χ$-bounded, as is the closure of $\mathcal{G}$ under the operations of substitution and gluing along a clique together.

math.CO↗
arXiv2011

The chromatic gap and its extremes

The {\em chromatic gap} is the difference between the chromatic number and the clique number of a graph. Here we investigate $\gap(n)$, the maximum chromatic gap over graphs on $n$ vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for $\gap(n)$ in terms of Ramsey numbers.

math.CO↗
arXiv2005

Coloring Artemis graphs

We consider the class A of graphs that contain no odd hole, no antihole, and no ``prism'' (a graph consisting of two disjoint triangles with three disjoint paths between them). We show that the coloring algorithm found by the second and fourth author can be implemented in time O(n^2m) for any graph in A with n vertices and m edges, thereby improving on the complexity proposed in the original paper.

cs.DM↗