125 research outputs found
A reconfigurations analogue of Brooks’ theorem.
Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless
G is a complete graph or a cycle with an odd number of vertices, or
k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike.
We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that
if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter,
if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2).
We complete this structural classification by settling the missing case:
if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2).
We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is
O(n 2) time solvable for k = 3,
PSPACE-complete for 4 ≤ k ≤ Δ(G),
O(n) time solvable for k = Δ(G) + 1,
O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)
A Reconfigurations Analogue of Brooks’ Theorem
Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2). We complete this structural classification by settling the missing case: if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is O(n 2) time solvable for k = 3, PSPACE-complete for 4 ≤ k ≤ Δ(G), O(n) time solvable for k = Δ(G) + 1, O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)
Topics in graph colouring and extremal graph theory
In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let be a connected graph with vertices and maximum degree . Let denote the graph with vertex set all proper -colourings of and two -colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that has a unique non-trivial component with diameter . This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours , of a component of the subgraph induced by vertices with colour or . Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all -colourings of a graph are Kempe equivalent unless is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees
Recognizing graphs close to bipartite graphs.
We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree
Token Jumping in minor-closed classes
Given two -independent sets and of a graph , one can ask if it
is possible to transform the one into the other in such a way that, at any
step, we replace one vertex of the current independent set by another while
keeping the property of being independent. Deciding this problem, known as the
Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar
graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by
if the input graph is -free.
We prove that the result of Ito et al. can be extended to any
-free graphs. In other words, if is a -free
graph, then it is possible to decide in FPT-time if can be transformed into
. As a by product, the TJ-reconfiguration problem is FPT in many well-known
classes of graphs such as any minor-free class
Partitioning a graph into degenerate subgraphs
Let be a connected graph with maximum degree distinct
from . Given integers and , is
said to be -partitionable if there exists a partition of
into sets~ such that is -degenerate for
. In this paper, we prove that we can find a -partition of in -time whenever and . This generalizes a result of Bonamy
et al. (MFCS, 2017) and can be viewed as an algorithmic extension of Brooks'
theorem and several results on vertex arboricity of graphs of bounded maximum
degree.
We also prove that deciding whether is -partitionable is
-complete for every and pairs of non-negative integers
such that and . This resolves an
open problem of Bonamy et al. (manuscript, 2017). Combined with results of
Borodin, Kostochka and Toft (\emph{Discrete Mathematics}, 2000), Yang and Yuan
(\emph{Discrete Mathematics}, 2006) and Wu, Yuan and Zhao (\emph{Journal of
Mathematical Study}, 1996), it also settles the complexity of deciding whether
a graph with bounded maximum degree can be partitioned into two subgraphs of
prescribed degeneracy.Comment: 16 pages; minor revisio
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