Recolouring planar graphs of girth at least five

Bartier, Valentin; Bousquet, Nicolás; Feghali, Carl; Heinrich, Marc; Moore, Benjamin; Pierron, Théo

doi:10.48550/arxiv.2112.00631

Cited by 2 publications

(4 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The existence of linear transformations between colorings received considerable attention in recent years, see e.g. [3,2,11,18]. Most of the works focus on the following question: how many colors are needed in order to guarantee the existence of a linear transformation between any pair of colorings of G?…”

Section: Our Resultsmentioning

confidence: 99%

Short and local transformations between ($Δ+1$)-colorings

Bousquet¹,

Feuilloley²,

Heinrich³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Recoloring a graph is about finding a sequence of proper colorings of this graph from an initial coloring σ to a target coloring η. Each pair of consecutive colorings must differ on exactly one vertex. The question becomes: is there a sequence of colorings from σ to η?In this paper, we focus on (∆ + 1)-colorings of graphs of maximum degree ∆. Feghali, Johnson and Paulusma proved that, if both colorings are non-frozen (i.e. we can change the color of a least one vertex), then a quadratic recoloring sequence always exists. We improve their result by proving that there actually exists a linear transformation.In addition, we prove that the core of our algorithm can be performed locally. Informally, if we start from a coloring where there is a set of well-spread nonfrozen vertices, then we can reach any other such coloring by recoloring only f (∆) independent sets one after another. Moreover these independent sets can be computed efficiently in the LOCAL model of distributed computing.

show abstract

Section: Our Resultsmentioning

confidence: 99%

Short and local transformations between ($Δ+1$)-colorings

Bousquet¹,

Feuilloley²,

Heinrich³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following Key Lemma is simple but powerful. It has been used implicitly in many papers, and first appeared explicitly in [1]. We phrase it in the slightly more general language of list-coloring, although the proof is identical.…”

Section: Preliminariesmentioning

confidence: 99%

“…(R1) Each 4-vertex gives 1/2 to each incident face. 1 (R2) Each 6 + -vertex gives 1 to each incident face.…”

Section: Preliminariesmentioning

confidence: 99%

“…Much work has focused on coloring reconfiguration. Recently, [1] completed the characterization of pairs (g, k) such that for all planar graphs G of girth at least g, every k-coloring of G can be transformed to every other. Similar problems have been studied for graphs of bounded maximum average degree [13], graphs of bounded treewidth [2], and particularly ddegenerate graphs [7].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

List-Recoloring of Sparse Graphs

Cranston¹

2022

Preprint

View full text Add to dashboard Cite

Fix a graph G, a list-assignment L for G, and L-colorings α and β. An L-recoloring sequence, starting from α, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper L-coloring. An L-recoloring sequence transforms α to β if its initial coloring is α and its final coloring is β. We prove there exists an L-recoloring sequence that transforms α to β and recolors each vertex at most a constant number of times if (i) G is triangle-free and planar and L is a 7-assignment, or (ii) mad(G) < 17/5 and L is a 6-assignment or (iii) mad(G) < 22/9 and L is a 4-assignment. Parts (i) and (ii) confirm conjectures of Dvořák and Feghali.

show abstract

Recolouring planar graphs of girth at least five

Cited by 2 publications

References 12 publications

Short and local transformations between ($Δ+1$)-colorings

Short and local transformations between ($Δ+1$)-colorings

List-Recoloring of Sparse Graphs

Contact Info

Product

Resources

About