Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces

Abstract

In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first linear-time algorithms to find such colorings.For a fixed surface Σ, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in Σ is 5-colorable and similarly in 2003 if a graph of girth at least five embedded in Σ is 3-colorable. Using the theory of hyperbolic families, the author and Thomas showed such algorithms exist for list-colorings. Around the same time, Dvořák and Kawarabayashi also provided such algorithms. Moreover, they gave an O(n O(g+1) )-time algorithm to find such colorings (if they exist) in n-vertex graphs where g is the Euler genus of the surface. Here we provide the first such algorithm which is fixed parameter tractable with genus as the parameter; indeed, we provide a linear-time algorithm to find such colorings.In 1988, Goldberg, Plotkin and Shannon provided a deterministic distributed algorithm for 7-coloring n-vertex planar graphs in O(log n) rounds. In 2018, Aboulker, Bonamy, Bousquet, and Esperet provided a deterministic distributed algorithm for 6-coloring n-vertex planar graphs in O(log 3 n) rounds. Their algorithm in fact works for 6-list-coloring. They also provided an O(log 3 n)round algorithm for 4-list-coloring triangle-free planar graphs. Chechik and Mukhtar independently obtained such algorithms for ordinary coloring in O(log n) rounds, which is best possible in terms of running time. Here we provide the first polylogarithmic deterministic distributed algorithms for 5-coloring n-vertex planar graphs and similarly for 3-coloring planar graphs of girth at least five. Indeed, these algorithms run in O(log n) rounds, work also for list-colorings, and even work on a fixed surface (assuming such a coloring exists).

Flexibility of planar graphs of girth at least six

“…It is relevant to mention that our proof of Theorem 2 uses the method of reducible configurations. The existence of a reducible configuration proof of 3-choosability of planar graphs of girth five has only been shown recently [5] and seems to require thousands of reducible configurations. Hence, we think to prove the part (c) of Problem 1, one would need to follow different ideas, possibly modifying Thomassen's 3-choosability proof [6].…”

Local Hadwiger's Conjecture

“…These theorems build on the ideas of the second author from [18]. In fact the algorithm is identical to the algorithm given in [18].…”

“…We recall the second author's definition of deletable subgraphs from [18]. Note for a vertex v in a graph G, we use d G (v) to denote the degree of v in G. Definition 1.15.…”

“…In 2019, Aboulker, Bonamy, Bousquet, and Esperet [1] gave a deterministic distributed algorithm for 6-list-colouring a planar graph G in O(log 3 (v(G))) rounds. The second author [18], using the theory of hyperbolic families, gave a deterministic distributed algorithm every fixed surface Σ for 5-list-colouring a graph G embedded in Σ in O(log(v(G))) rounds.…”

Flexibility of triangle‐free planar graphs

“…In particular, a request is 1‐satisfiable if and only if the precoloring given by extends to an ‐coloring of . The corresponding precoloring extension problem has been studied in a number of contexts: as a tool to deal with small cuts in the considered graph by coloring one part of the graph recursively and then extending the corresponding precoloring of the cut vertices to the other part [2,5,13], as a way to show that a graph has many different colorings [6,12], or from the algorithmic complexity perspective [3,9]. In planar graphs, it is known that any precoloring of a set of vertices at distance at least three from one another extends when at least 5 colors are used [1] and for sufficiently distant vertices this holds also in the list coloring setting [7]; on the other hand, a precoloring of two arbitrarily distant vertices of a planar graph does not necessarily extend to a 4‐coloring [10].…”