Generalised colouring numbers of planar graphs
Daniel Quiroz
Joint work with Jan van den Heuvel, Patrice Ossona de Mendez,
Roman Rabinovich, Sebastian Siebertz
London School of Economics and Political Science
LSE, February, 2016
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Generalised colouring numbers
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Colouring number
u, v ∈ SL,1 (y )
Let G be a graph.
Let L be an linear ordering of V (G ).
u
v
y
w
We denote by SL,1 (y ) the set of neighbours of y that are before y in L.
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Generalised colouring numbers
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Colouring number
u, v ∈ SL,1 (y )
Let G be a graph.
Let L be an linear ordering of V (G ).
u
v
y
w
We denote by SL,1 (y ) the set of neighbours of y that are before y in L.
The colouring number of G is
col(G ) = 1 + min max |SL,1 (y )|.
L y ∈V (G )
Clearly, χ(G ) ≤ col(G ).
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Generalised colouring numbers
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Strong r -reachability and the strong r -colouring number
First introduced by Kierstead and Yang (2003).
Let L be an ordering of V (G ).
A vertex u is strong r -reachable from y if u <L y and there exists an
uy -path P of length at most r , such that every internal vertex z of P
satisfies y <L z.
u
y
v
u ∈ SL,3 (y )
Daniel Quiroz (LSE)
y
v ∈ SL,3 (y )
Generalised colouring numbers
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Strong r -reachability and the strong r -colouring number
First introduced by Kierstead and Yang (2003).
Let L be an ordering of V (G ).
A vertex u is strong r -reachable from y if u <L y and there exists an
uy -path P of length at most r , such that every internal vertex z of P
satisfies y <L z.
Let SL,r (y ) be the set of vertices which are strong r -reachable from y given
the ordering L, and define the strong r -colouring number:
colr (G ) = 1 + min max |SL,r (y )|.
L y ∈V (G )
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Generalised colouring numbers
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Weak r -reachability and weak r -colouring number
A vertex u is weak r -reachable from y if u <L y and there exists an
uy -path P of length at most r , such that every internal vertex z of P
satisfies u <L z.
u
y
v
u ∈ WL,3 (y )
Daniel Quiroz (LSE)
y
v ∈ WL,3 (y )
Generalised colouring numbers
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Weak r -reachability and weak r -colouring number
A vertex u is weak r -reachable from y if u <L y and there exists an
uy -path P of length at most r , such that every internal vertex z of P
satisfies u <L z.
u
y
v
u ∈ WL,3 (y )
y
v ∈ WL,3 (y )
Let WL,r (y ) be the set of vertices which are weakly r -reachable from y
given the ordering L, and define the weak r -colouring number:
wcolr (G ) = 1 + min max |WL,r (y )|.
L y ∈V (G )
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Generalised colouring numbers
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A rough inequality
Lemma (Kierstead and Yang, 2003)
Every graph G satisfies
wcolr (G ) ≤ (colr (G ))r .
Proof: Partition each “weak path” into “strong paths”.
u
y
u ∈ WL,5 (y )
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Nice generalisations but. . . why study these numbers?
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Generalised colouring numbers
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Nice generalisations but. . . why study these numbers?
Proposition (Grohe et al., 2015)
col(G ) = col1 (G ) ≤ col2 (G ) ≤ · · · ≤ col∞ (G ) = tw(G ) + 1
Proposition (Nešetřil and Ossona de Mendez, 2012)
col(G ) = wcol1 (G ) ≤ wcol2 (G ) ≤ · · · ≤ wcol∞ (G ) = td(G )
Daniel Quiroz (LSE)
Generalised colouring numbers
LSE, February, 2016
7 / 19
Nice generalisations but. . . why study these numbers?
Proposition (Grohe et al., 2015)
col(G ) = col1 (G ) ≤ col2 (G ) ≤ · · · ≤ col∞ (G ) = tw(G ) + 1
Proposition (Nešetřil and Ossona de Mendez, 2012)
col(G ) = wcol1 (G ) ≤ wcol2 (G ) ≤ · · · ≤ wcol∞ (G ) = td(G )
• Applications in Ramsey theory (arrangeablity), graph structures (classes
with bounded expansion), different types of colourings. . .
Daniel Quiroz (LSE)
Generalised colouring numbers
LSE, February, 2016
7 / 19
Nice generalisations but. . . why study these numbers?
Proposition (Grohe et al., 2015)
col(G ) = col1 (G ) ≤ col2 (G ) ≤ · · · ≤ col∞ (G ) = tw(G ) + 1
Proposition (Nešetřil and Ossona de Mendez, 2012)
col(G ) = wcol1 (G ) ≤ wcol2 (G ) ≤ · · · ≤ wcol∞ (G ) = td(G )
• Applications in Ramsey theory (arrangeablity), graph structures (classes
with bounded expansion), different types of colourings. . .
• Personal motivation:
Theorem (Van den Heuvel, Kierstead and Q., 2016+)
Let G be a graph and let k be a non-negative integer.
Then χ(G [\2k+1] ) ≤ wcol4k+1 (G ).
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Generalised colouring numbers
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Upper bounds when excluding Kt as a minor
• Zhu (2007) proved that if a graph G excludes the complete graph Kt as
a minor then
colr (G ) ≤ 1 + qr ,
2
where q1 = t − 1, and qi+1 = q1 · qi2i for i ≥ 1.
• Grohe et al. (2015) improved these result by showing that
colr (G ) ≤ (crt)r .
• From these we could obtain bounds on wcolr (G ) by using that
wcolr (G ) ≤ (colr (G ))r .
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Generalised colouring numbers
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Upper bounds on colr (G ) when G excludes Kt as a minor
Theorem (Van den Heuvel et al., 2016+)
Every graph G that excludes the complete graph Kt as a minor satisfies
t −1
colr (G ) ≤
· (2r + 1).
2
It can be shown that the r × r grid Gr ×r satisfies colr (Gr ×r ) ∈ Ω(r ).
Of course, Gr ×r excludes K5 , so linear in r is best possible for r ≥ 5.
Daniel Quiroz (LSE)
Generalised colouring numbers
LSE, February, 2016
9 / 19
Upper bounds on colr (G ) when G excludes Kt as a minor
Theorem (Van den Heuvel et al., 2016+)
Every graph G that excludes the complete graph Kt as a minor satisfies
t −1
colr (G ) ≤
· (2r + 1).
2
It can be shown that the r × r grid Gr ×r satisfies colr (Gr ×r ) ∈ Ω(r ).
Of course, Gr ×r excludes K5 , so linear in r is best possible for r ≥ 5.
Theorem (Van den Heuvel et al., 2016+)
Every planar graph G satisfies
colr (G ) ≤ 5r + 1.
Optimal for r = 1.
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Generalised colouring numbers
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Upper bounds on wcolr (G ) when G excludes Kt as a minor
Theorem (Van den Heuvel et al., 2016+)
Every graph G that excludes the complete graph Kt as a minor satisfies
r +t −2
wcolr (G ) ≤
· (t − 3)(2r + 1) ∈ O(r t−1 ).
t −2
Grohe et al. (2015) showed that for every t there is a graph Gt,r which
excludes Kt as a minor and satisfies
r +t −2
wcol(Gt,r ) =
∈ Ω(r t−2 ).
t −2
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Generalised colouring numbers
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Upper bounds on wcolr (G ) when G excludes Kt as a minor
Theorem (Van den Heuvel et al., 2016+)
Every graph G that excludes the complete graph Kt as a minor satisfies
r +t −2
wcolr (G ) ≤
· (t − 3)(2r + 1) ∈ O(r t−1 ).
t −2
Grohe et al. (2015) showed that for every t there is a graph Gt,r which
excludes Kt as a minor and satisfies
r +t −2
wcol(Gt,r ) =
∈ Ω(r t−2 ).
t −2
Theorem (Van den Heuvel et al., 2016+)
Every planar graph G satisfies
r +2
wcolr (G ) ≤
· (2r + 1) ∈ O(r 3 ).
2
Daniel Quiroz (LSE)
Generalised colouring numbers
LSE, February, 2016
10 / 19
Isometric paths
Let G be a graph. We call a path P in G an isometric path if P is a
shortest path between its endpoints.
Lemma
Let y be a vertex of a graph G , and let P be an isometric path in G . Then
P contains at most 2r + 1 vertices of the closed r -neighbourhood of v :
|Nr [y ] ∩ V (P)| ≤ 2r + 1.
Where Nr [y ] is the set of vertices with distance at most r from y , including
y.
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Generalised colouring numbers
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Isometric paths
Let G be a graph. We call a path P in G an isometric path if P is a
shortest path between its endpoints.
Lemma
Let y be a vertex of a graph G , and let P be an isometric path in G . Then
P contains at most 2r + 1 vertices of the closed r -neighbourhood of v :
|Nr [y ] ∩ V (P)| ≤ 2r + 1.
Since for every ordering L, and every r and y we have
SL,r (y ) ⊆ WL,r (y ) ⊆ Nr [y ]
we obtain
|SL,r (y ) ∩ V (P)| ≤ |WL,r (y ) ∩ V (P)| ≤ 2r + 1.
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Generalised colouring numbers
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Isometric paths decompositions
An isometric paths decomposition of a graph G is aSsequence
P = (P1 , . . . , P` ) of vertex-disjoint paths such thatS `i=1 V (Pi ) = V (G ),
and such that each Pi is an isometric path in G − 1≤j≤i−1 V (Pj ).
S
Let i ≥ 1 and let C be a component of G − 1≤j≤i V (Pj ). The separating
number of C is the number s of (distinct) paths Q1 , . . . , Qs ∈ {P1 , . . . , Pi }
such that C has a neighbour in each Qj .
The width of a decomposition P is the maximum separating S
number,
maximised over all i, 1 ≤ i ≤ `, and all components of G − 1≤j≤i V (Pj ).
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Generalised colouring numbers
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Isometric paths and tree-width
Lemma
Let G be a graph, and let P = (P1 , . . . , P` ) be an isometric paths
decomposition of G of width k. By contracting each path Pi to a single
vertex, we obtain a graph H with tree-width at most k.
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Generalised colouring numbers
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Isometric paths and tree-width
Lemma
Let G be a graph, and let P = (P1 , . . . , P` ) be an isometric paths
decomposition of G of width k. By contracting each path Pi to a single
vertex, we obtain a graph H with tree-width at most k.
Theorem (Grohe et al., 2015)
Every graph H with tree-width at most k satisfies
r +k
wcolr (H) ≤
k
This tells us that there is an ordering
of the paths, so that each path
weakly r -reaches at most r +k
−
1
other
paths.
k
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Generalised colouring numbers
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Isometric paths decomposition and wcolr (G )
Lemma
Every maximal planar graph admits an isometric path decomposition of
width 2.
Theorem (Van den Heuvel et al., 2016+)
Every planar graph G satisfies
r +2
wcolr (G ) ≤
· (2r + 1) ∈ O(r 3 ).
2
Adding edges (to make G maximal) cannot decrease wcolr (G ).
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Generalised colouring numbers
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Isometric paths decomposition for planar graphs
Lemma
Every maximal planar graph admits an isometric path decomposition of
width 2.
Proof: Let |V (G )| ≥ 3. The outer face of a maximal planar graph is a
triangle. Pick P1 as an edge of that triangle and P2 as the remaining
vertex. We inductively construct a decomposition where each component
C of G − ∪1≤j≤i Pj , for all i ≥ 2, has neighbours in exactly two paths of
P1 , . . . , Pi .
Pa
e2
v2
C
e1
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v1
Pb
Generalised colouring numbers
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Isometric path decompositions and colr (G )
From the same decomposition we can deduce that colr (G ) ≤ 5r + 3.
Pa
e2
v2
C
e1
v1
Pb
With a different type of decomposition and careful analysis we can prove
Theorem (Van den Heuvel et al., 2016+)
Every planar graph G satisfies
colr (G ) ≤ 5r + 1.
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Generalised colouring numbers
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Graphs with genus g
Theorem (Van den Heuvel et al., 2016+)
Every graph G with genus g satisfies
r +2
wcolr (G ) ≤ 2g +
· (2r + 1).
2
Proof: For a graph of genus g > 0, there exists a non-separating cycle C
that consists of two isometric paths such that G − C has genus g − 1.
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Graphs with genus g
Theorem (Van den Heuvel et al., 2016+)
Every graph G with genus g satisfies
r +2
wcolr (G ) ≤ 2g +
· (2r + 1).
2
Proof: For a graph of genus g > 0, there exists a non-separating cycle C
that consists of two isometric paths such that G − C has genus g − 1.
Similarly we can prove
Theorem (Van den Heuvel et al., 2016+)
Every graph G with genus g satisfies
colr (G ) ≤ 2g (2r + 1) + 5r + 1.
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Thank you for listening!
¡Gracias!
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