A decomposition for total-coloring graphs of maximum degree 3 - - PowerPoint PPT Presentation
A decomposition for total-coloring graphs of maximum degree 3 Cologne-Twente Workshop 2008 Raphael Machado and Celina M. H. de Figueiredo Gargnano. 14/05/2008. General concepts We work with simple graphs G= ( V ( G ) ,E ( G )) V ( G )
We work with simple graphs G=(V(G),E(G))
Set of elements of G: S(G)=V(G)∪E(G)
Vertices u,v∈V(G) are adjacent if uv∈E(G) Edges e1,e2∈E(G) are adjacent if they have a
Vertex u∈V(G) and edge e∈E(G) are incident
Open neighborhood: AdjG(u):={v∈V(G)|uv∈E(G)} Closed neighborhood: NG(u):= AdjG(u)∪{u} Degree: degG(u):=|AdjG(u)| Maximum degree: ∆(G)
Associated to conflict models Related elements (incident or adjacent)
Three classical models
Is an association of colors to the elements of a graph Incident or adjacent elements receive distinct colors k-total-coloring: a total-coloring that uses k colors k-total-colorable graph: it can be colored with k colors Total chromatic number χT(G): least number of colors
Observe that χT(G) ≥ ∆(G) + 1 Total Coloring Conjecture: χT(G) ≤ ∆(G) + 2 Classification problem:
It is NP-complete to determine if a graph is
Gmxn is a grid if
P2 and C4 are Type 2 All other grids are Type 1
Arbitrary subgraphs of grids
Total chromatic number determined for ∆ = 0,
Open problem for ∆=3.
Development of a decomposition method for
Classification of partial grids with maximum
(Some recent results in series-parallel graphs total-coloring)
As a first step we formalize a result that allows
If G is a graph such that all of its biconnected
A cut of a graph G is a set of vertices whose
If C⊆V(G) is a cut whose exclusion defines the
A K2-cut is a cut {u,v} such that u and v are
The K2-cut-free components of G are defined
The set {u,v} is a frontier-candidate if u and v are
Let {u,v} be a frontier candidate and denote u’≠v and
We say that a coloring satisfies the frontier condition
A frontier-coloring is a coloring that satisfies
Reference vertex
Not a frontier- coloring
Reference vertex for G1 Reference vertex for G2
Consider a biconnected graph G of maximum
Suppose each K2-cut-free component of G has two
In this case, G is 4-total-colorable.
If G is a biconnected graph of maximum
The above result allows us to 4-total-color G
Sketch of proof
A result: 8-chordal partial-grids with ∆=3 are
A graph is a SP-graph if it has no subgraph
Subclass of planar, the {K5,K3,3}-free graphs Superclass of outerplanar, the {K4,K2,3}-free
Every SP-graph of maximum degree ∆>3 is
The total chromatic number of graphs of
The only open case is ∆ = 3 We can apply our technique for subclasses with
A result: 6-chordal partial-grids with ∆=3 are
Our results
Future goals