A note on the flip distance between non-crossing spanning trees

We consider spanning trees of $n$ points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive $2n-\Omega(1)$ upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a $2n-\Omega(\log n)$ bound. We improve their result with a $2n-\Omega(\sqrt{n})$ upper bound, and we strengthen and shorten the proofs of several of their results

A note on deterministic zombies

"Zombies and Survivor" is a variant of the well-studied game of "Cops and Robber" where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path if multiple options are available. The zombie number, like the cop number, of a graph is the minimum number of zombies, or cops, required to capture the survivor. In this short note, we solve a question by Fitzpatrick et al., proving that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers. We also give a simple graph family with cop number $2$ and an arbitrarily large zombie number.Comment: 4 page

A note on connected greedy edge colouring

Following a given ordering of the edges of a graph $G$, the greedy edge colouring procedure assigns to each edge the smallest available colour. The minimum number of colours thus involved is the chromatic index $\chi'(G)$, and the maximum is the so-called Grundy chromatic index. Here, we are interested in the restricted case where the ordering of the edges builds the graph in a connected fashion. Let $\chi_c'(G)$ be the minimum number of colours involved following such an ordering. We show that it is NP-hard to determine whether $\chi_c'(G)>\chi'(G)$. We prove that $\chi'(G)=\chi_c'(G)$ if $G$ is bipartite, and that $\chi_c'(G)\leq 4$ if $G$ is subcubic.Comment: Comments welcome, 12 page

Coloration et recoloration de graphes

Cette thèse s'inscrit dans le cadre de la théorie des graphes, et plus précisément dans le cadre de la coloration de graphes avec une attention particulière portée à la coloration d'arêtes, et la reconfiguration de colorations. Dans cette thèse, nous étudions principalement les changements de Kempe, un outil de transformation locale d'une coloration en une autre coloration. Ce concept est une idée clef de la preuve du théorème des 4 couleurs. Nous donnons un aperçu de l'histoire de cet outil technique, décrivons la manière dont il est devenu l'un des outils les plus prolifiques quant aux questions de colorations de graphes, et présentons des questions, s'inscrivant dans le cadre plus général de la reconfiguration combinatoire, issues de ce concept.Nous présentons ensuite nos résultats sur la coloration gloutonne d'arêtes et la reconfiguration de coloration de sommets pour les graphes sans K_t comme mineurs. En ce qui concerne la reconfiguration de coloration d'arêtes, nous prouvons en particulier que toutes les (chi'(G)+1)-colorations sont Kempe-équivalentes entre elles (i.e. qu'il est possible de transformer n'importe quelle coloration en n'importe quelle autre coloration en utilisant uniquement des changements de Kempe), prouvant ainsi une conjecture de Vizing de 1965. Nous présentons enfin notre travail sur la coloration de sommets de graphes signés, et sur la coloration d'arêtes de graphes planaires sans triangle de degré maximum 4.This thesis falls within the field of graph theory, and more precisely of graph coloring with a focus on coloring reconfiguration and edge-coloring. In this thesis, we mainly study the Kempe swaps, a tool to locally transform a coloring into another coloring. This concept is one of the key ideas of the proof of the 4-color theorem. We first give an overview of the history of this tool, present how it became one of the most fruitful tool regarding graph coloring questions, and introduce questions that fall within the more general field of combinatorial reconfiguration that emerged from this concept.We then present our result on greedy edge-coloring and vertex-coloring reconfiguration of K_t-minor free graphs. Regarding edge-coloring reconfiguration, we prove in particular that all (chi'(G)+1)-colorings are Kempe equivalent (i.e. one can transform any coloring into any other coloring using only Kempe swaps), thus proving a conjecture of Vizing of 1965. We finally present our work on coloring of signed planar graphs, and on edge-coloring of triangle-free planar graphs of maximum degree 4

Coloration et recoloration de graphes

This thesis falls within the field of graph theory, and more precisely of graph coloring with a focus on coloring reconfiguration and edge-coloring. In this thesis, we mainly study the Kempe swaps, a tool to locally transform a coloring into another coloring. This concept is one of the key ideas of the proof of the 4-color theorem. We first give an overview of the history of this tool, present how it became one of the most fruitful tool regarding graph coloring questions, and introduce questions that fall within the more general field of combinatorial reconfiguration that emerged from this concept.We then present our result on greedy edge-coloring and vertex-coloring reconfiguration of K_t-minor free graphs. Regarding edge-coloring reconfiguration, we prove in particular that all (chi'(G)+1)-colorings are Kempe equivalent (i.e. one can transform any coloring into any other coloring using only Kempe swaps), thus proving a conjecture of Vizing of 1965. We finally present our work on coloring of signed planar graphs, and on edge-coloring of triangle-free planar graphs of maximum degree 4.Cette thèse s'inscrit dans le cadre de la théorie des graphes, et plus précisément dans le cadre de la coloration de graphes avec une attention particulière portée à la coloration d'arêtes, et la reconfiguration de colorations. Dans cette thèse, nous étudions principalement les changements de Kempe, un outil de transformation locale d'une coloration en une autre coloration. Ce concept est une idée clef de la preuve du théorème des 4 couleurs. Nous donnons un aperçu de l'histoire de cet outil technique, décrivons la manière dont il est devenu l'un des outils les plus prolifiques quant aux questions de colorations de graphes, et présentons des questions, s'inscrivant dans le cadre plus général de la reconfiguration combinatoire, issues de ce concept.Nous présentons ensuite nos résultats sur la coloration gloutonne d'arêtes et la reconfiguration de coloration de sommets pour les graphes sans K_t comme mineurs. En ce qui concerne la reconfiguration de coloration d'arêtes, nous prouvons en particulier que toutes les (chi'(G)+1)-colorations sont Kempe-équivalentes entre elles (i.e. qu'il est possible de transformer n'importe quelle coloration en n'importe quelle autre coloration en utilisant uniquement des changements de Kempe), prouvant ainsi une conjecture de Vizing de 1965. Nous présentons enfin notre travail sur la coloration de sommets de graphes signés, et sur la coloration d'arêtes de graphes planaires sans triangle de degré maximum 4

Circular (4 − \epsilon)-coloring of some classes of signed graphs

A circular r-coloring of a signed graph (G, σ) is an assignment φ of points of a circle C r of circumference r to the vertices of (G, σ) such that for each positive edge uv of (G, σ) the distance of φ(v) and φ(v) is at least 1 and for each negative edge uv the distance of φ(u) from the antipodal of φ(v) is at least 1. The circular chromatic number of (G, σ), denoted χ c (G, σ), is the infimum of r such that (G, σ) admits a circular r-coloring. This notion is recently defined by Naserasr, Wang, and Zhu who, among other results, proved that for any signed d-degenerate simple graphĜ we have χ c (Ĝ) ≤ 2d. For d ≥ 3, examples of signed d-degenerate simple graphs of circular chromatic number 2d are provided. But for d = 2 only examples of signed 2-degenerate simple graphs of circular chromatic number close enough to 4 are given, noting that these examples are also signed bipartite planar graphs. In this work we first observe the following restatement of the 4-color theorem: If (G, σ) is a signed bipartite planar simple graph where vertices of one part are all of degree 2, then χ c (G, σ) ≤ 16 5. Motivated by this observation, we provide an improved upper bound of 4 − 2 n+1 2 for the circular chromatic number of a signed 2-degenerate simple graph on n vertices and an improved upper bound of 4 − 4 n+2 2 for the circular chromatic number of a signed bipartite planar simple graph on n vertices. We then show that each of the bounds is tight for any value of n ≥ 4

A Note on the Flip Distance between Non-crossing Spanning Trees

International audienceWe consider spanning trees of n points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree. Given two trees, we investigate the minimum number of flips required to transform one into the other. The naive 2n - Omega(1) upper bound stood for 25 years until a recent breakthrough from Aichholzer et al. yielding a 2n - Ω(log n) bound. We improve their result with a 2n - Ω(√n) upper bound, and we strengthen and shorten the proofs of several of their results

On a recolouring version of Hadwiger's conjecture

We prove that for any ε>0, for any large enough t, there is a graph G that admits no Kt-minor but admits a (3/2−ε)t-colouring that is "frozen" with respect to Kempe changes, i.e. any two colour classes induce a connected component. This disproves three conjectures of Las Vergnas and Meyniel from 1981

On Vizing's edge colouring question

International audienceSoon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs

A note on deterministic zombies

International audienceZombies and Survivor is a variant of the well-studied game of Cops and Robbers where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path if multiple options are available. Similar to the cop number, the zombie number of a graph is the minimum number of zombies required to capture the survivor. In this short note, we solve a question by Fitzpatrick et al., proving that the zombie number of the Cartesian product of two graphs is at most the sum of their zombie numbers. We also give a simple graph family with cop number 2 and an arbitrarily large zombie number