Enumerating the edge-colourings and total colourings of a regular graph

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HAL Id: lirmm-00811571

https://hal-lirmm.ccsd.cnrs.fr/lirmm-00811571

Submitted on 10 Apr 2013

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Stéphane Bessy, Frédéric Havet

To cite this version:

Stéphane Bessy, Frédéric Havet. Enumerating the edge-colourings and total colourings of a regu-lar graph. 2012 Workshop on Graph Theory and Combinatorics, Aug 2012, National Sun Yat-sen University, Kaohsiung, Taiwan. �lirmm-00811571�

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Enumerating the edge-colourings of a regular graph

Stéphane Bessy and Frédéric Havet

Assistant Professor Researcher

LIRMM, UM2, Montpellier CNRS, Sophia-Antipolis FRANCE

2012 Workshop on Graph Theory and Combinatorics

Deparment of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan.

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1 Introduction Colorings

Algorithmic problems Our results

2 Enumerating the 3-edge colorings of a cubic graph

The 3-edge colorings of a 3-regular graph Turning the proof into algorithm

3 Extensions: k-edge colorings and the total colorings

k-edge colorings

A more precise bounds for the 3-edge coloring Total coloring

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Introduction

1 Introduction

Colorings

2 Enumerating the 3-edge colorings of a cubic graph The 3-edge colorings of a 3-regular graph Turning the proof into algorithm

k-edge colorings

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Definition (k-vertex coloring)

A k-vertex coloring of a (non oriented) graph G = (V , E ) is a function c : V → {1, . . . , k} such that uv ∈ E implies c(u) 6= c(v ).

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Vertex Coloring

b c

f a

e d

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b c

f a

e d

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Vertex Coloring

b c

f a

e d

Definition (chromatic number)

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Definition (k-edge coloring)

A k-edge coloring of a graph G = (V , E ) is a function c : E → {1, . . . , k} such that uv ∈ E and uw ∈ E , with v 6= w implies c(uv ) 6= c(uw ).

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Edge Coloring

b c

f a

e d

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b c

f a

e d

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Edge Coloring

b c

f a

e d

Definition (chromatic index)

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The line graph L(G ) of G :

- The vertex set of L(G ) is the edge set of G

- Two edges e and f of G are adjacent in L(G ) if there are incident in G G L(G ) ab af bf bc ef ce cd de a b c d e f

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Edge Coloring

The line graph L(G ) of G :

- The vertex set of L(G ) is the edge set of G

- Two edges e and f of G are adjacent in L(G ) if there are incident in G G L(G ) ab af bf bc ef ce cd de a b c d e f χ′(G ) = χ(L(G ))

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Definition (k-total coloring)

A k-total coloring of G = (V , E ) is c : V ∪ E → {1, . . . , k} such that: - uv ∈ E implies c(u) 6= c(v )

- uv ∈ E and uw ∈ E , with v 6= w implies c(uv ) 6= c(uw ) - uv ∈ E implies c(uv ) 6= c(u)

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Total Coloring

b c

f a

e d

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b c

f a

e d

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Total Coloring

b c

f a

e d

Definition (total chromatic number)

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Famous bounds

Let ∆(G ) denotes the maximum degree of G .

Brook’s Theorem (1941): χ(G ) ≤ ∆(G ) if G is not a complete graph or an odd cycle.

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Famous bounds

Vizing’s Theorem (1964): ∆(G ) ≤ χ′_{(G ) ≤ ∆(G ) + 1}

Total coloring conjecture (Behzad, Vizing, ∼1964):

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k-vertex/edge/total coloring: - Input: a graph G = (V , E )

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Algorithmic problems

- Output: Does G admit a k-vertex/edge/total coloring? k-vertex coloring is NP-complete for every fixed k ≥ 3 (M.R. Garey and D.S. Johnson, 1979).

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k-edge coloring is NP-complete for every fixed k ≥ 3 (I. Holyer, 1981).

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Algorithmic problems

k-total coloring is NP-complete for every fixed k ≥ 4 (A. S´anchez-Arroyo, 1989).

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Enum-k-vertex/edge/total coloring: - Input: a graph G = (V , E )

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Algorithmic problems

Enum-k-vertex/edge/total coloring: - Input: a graph G = (V , E )

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Algorithmic point of view

What to do with an NP-complete problem?

approximation algorithm: find in poly-time a result which is not worst than c time the optimal.

randomized algorithm: find a result which is good w.h.p. or/and with running time correct w.h.p.

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...

parameterized algorithm: find an algorithm with running time f (k).poly (n) for some parameter k.

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Algorithmic point of view

...

parameterized algorithm: find an algorithm with running time f (k).poly (n) for some parameter k.

exact (exponential) algorithms: find an algorithm with running time O⋆(cn_{) (= O(P(n)c}n₎₎

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Enum-3-edge coloring:

- Input: a graph G = (V , E ) with ∆(G ) ≤ 3.

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Examples of exact algorithms

- Output: Enumerate all the 3-vertex/edge/total coloring of G . The exhaustive algorithm:

- Try every possible colorings of the edge set and decide which colorings are proper.

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Time analysis:

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Examples of exact algorithms

Time analysis:

O⋆₍₃m_{) = O}⋆₍₃3₂n_{) = O}⋆_(5.1962n₎

Usual methods to improve: branching algorithms and dynamic programming.

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Enum-3-edge coloring: a ’naive’ branching algorithms a b c d ab bc ad ac

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Examples of exact algorithms

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Examples of exact algorithms

Time analysis: O⋆(2m_{) = O}⋆₍₂3₂n_{) = O}⋆_(2.8284n₎ _{min. nb of leaves}

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Previous work: exact algorithms

What is the gain on the running time? In the same environment:

t = O⋆(an1_{) = O}⋆_(bn2_{) with a < b} We obtain: n1 ∼ log b_{log a}n2

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What is the gain on the running time? In the same environment:

t = O⋆(an1_{) = O}⋆_(bn2_{) with a < b} We obtain: n1 ∼ log b_{log a}n2

A multiplicative factor on the instance size of log b_{log a} > 1 Here: log 3

3 2

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Previous work: exact algorithms

Main results in exact algorithms to color a graph:

3-vertex coloring in O⋆(1.3289n_{) (R. Beigel and D. Eppstein,}

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2005)

Minimum vertex coloring in O⋆(2n_{) (A. Bj¨}_{oklund, T. Husfeld}

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Previous work: exact algorithms

2005)

and M. Koivisto, 2006)

Enum 3-vertex coloring in O⋆(1.6259n_{) (F. Fomin, S. Gasper}

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2005)

and S. Saurabh, 2007)

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Previous work: exact algorithms

2005)

3-edge coloring in O⋆_(1.344n_{) (L. Kowalik, 2009)}

Enum-3-edge coloring in O⋆(1.5423n_{) (P.A. Golovatch,}

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2005)

D. Kratsch and J.F. Couturier, 2010)

Enum-4-total coloring in O⋆(3.0845n_{) (P.A. Golovatch,}

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Previous work: exact algorithms

2005)

D. Kratsch and J.F. Couturier, 2010)

Enum-4-total coloring in O⋆(3.0845n_{) (P.A. Golovatch,}

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Motivations:

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Our results

Motivations:

a useful technique for edge coloring 3-edge coloring more interesting

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Motivations:

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Our results

Motivations:

Gap: Ω(1.2822n_{) ≤ Enum-3-edge coloring ≤ O}⋆_(1.5423n₎

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Motivations:

? ≤ Enum-4-total coloring ≤ O⋆_(3.0845n₎

Our results:

Enum-3-edge coloring in O⋆(1.4142n_{) and a sharp example}

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Our results

Motivations:

Our results:

(multi-graph). (factor on data size: 1.25)

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Motivations:

Our results:

Enum-4-total coloring in O⋆_(2.8285n_{). Example in Ω(2.3784}n_).

Enum-k-edge coloring in O⋆((k − 1!)n2) and a sharp example

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Our results

Motivations:

Our results:

Enum-k-edge coloring in O⋆((k − 1!)n2) and a sharp example

(multi-graph).

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2 Enumerating the 3-edge colorings of a cubic graph

The 3-edge colorings of a 3-regular graph Turning the proof into algorithm

k-edge colorings

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Enum-3-edge coloring

We denote by ck(G ) the number of k-edge-colorings of a graph G .

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Here, we want to compute c3(G ), G being a sub-cubic (multiple) graph.

We can assume that G is connected. G1

G2

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Enum-3-edge coloring

Here, we want to compute c3(G ), G being a sub-cubic (multiple) graph.

We can assume that G is connected. G1

G2

c3(G ) = c3(G1) · c3(G2)

We can assume that G is 2-(vertex) connected.

G1

G2

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Lemma

Let Cn be the cycle of length n.

c3(Cn) =

2n_{+ 2, if n is even,}

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Number of 3-edge colorings of 3-regular graphs

Lemma

c3(Cn) =

2n_{− 2, if n is odd.}

Proof: By induction on n. True for n = 2 and n = 3. v2 v1 vn vn−1 v3 vn−2 v2 vn−1 v3 vn−2 c3(Cn) = 2.c3(Cn−2)+

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Lemma

c3(Cn) =

2n_{− 2, if n is odd.}

Proof: By induction on n. True for n = 2 and n = 3. v2 v1 vn vn−1 v3 vn−2 v2 vn−1 v3 vn−2 v1

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Number of 3-edge colorings of 3-regular graphs

We denote by ni the number of vertices of degree i in G .

Theorem

Let G be a 2-connected subcubic graph. Then c3(G ) ≤ 3 · 2n−

n3 2.

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We denote by ni the number of vertices of degree i in G .

Theorem

n3 2.

Proof: If G is a cycle, it is true.

Otherwise, let v1 and vn be two vertices of degree 3 and consider

v1, . . . , vn an st-ordering of G :

for all 1 < i < n, d(vi)_{v1,...,vi −1}≥ 1 and d(vi){vi+1,...,vn} ≥ 1. (always exists: Lempel et al.,1967)

v3 v2

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Number of 3-edge colorings of 3-regular graphs

Theorem

n3 2. Proof: v3 v2 v1 v4 v5 v6

Orient from left to right.

A+_{= {v : d}+_{(v ) = 2 and d}−_{(v ) = 1}}

A−= {v : d+(v ) = 1 and d−(v ) = 2}

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Theorem

n3 2. Proof: v3 v2 v1 v4 v5 v6

A+_{= {v : d}+_{(v ) = 2 and d}−_{(v ) = 1}}

A−= {v : d+(v ) = 1 and d−(v ) = 2}

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Number of 3-edge colorings of 3-regular graphs

Theorem

n3 2.

Proof: We have |A+| = |A−| = (n3− 2)/2

Denote by ci the number of partial 3-edge colorings of arcs with tail in

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Theorem

n3 2.

Proof: We have |A+| = |A−| = (n3− 2)/2

{v1, . . . , vi}. c1 = 6 if vi ∈ A− vi ci ≤ ci−1 if vi ∈ A+ vi ci ≤ 2ci−1 if d+(vi) = d−(vi) = 1 vi ci ≤ 2ci₋₁

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Number of 3-edge colorings of 3-regular graphs

Theorem

n3 2.

Proof: We have |A+| = |A−| = (n3− 2)/2

{v1, . . . , vi}. c1 = 6 if vi ∈ A− vi ci ≤ ci−1 if vi ∈ A+ vi ci ≤ 2ci−1 if d+(vi) = d−(vi) = 1 vi ci ≤ 2ci₋₁ c3(G ) ≤ 6 · 2(n−2)−(n3−2)/2= 3 · 2n− n3 2

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Theorem

n3 2.

Corollary

Let G be a cubic graph, then c3(G ) ≤ 3 · 2

n 2.

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Enum-3-edge coloring:

Corollary (Solving Enum-3-edge coloring:)

There exists a branching algorithm with running time O⋆₍₂n₂_{) =}

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There exists a branching algorithm with running time O⋆₍₂n₂_{) =}

O⋆(1.4143n_{) and polynomial space to solve Enum-3-edge coloring.}

Proof:

If ∆ ≥ 4 answer 0.

If the graph is not connected enough, divide the instance. Otherwise, run a branching algorithm.

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k-edge colorings

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Number of k-edge colorings of k-regular graphs

We denote by ck the number of k-edge colorings of G .

Theorem

Let G be a k-regular graph. Then ck(G ) ≤ k · (k − 1!)

n 2.

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Theorem

n 2.

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Number of k-edge colorings of k-regular graphs

Theorem

n 2.

Proof: G is 2-connected,consider v1, . . . , vn an st-ordering of G :

v3 v2

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Theorem

n 2.

Proof: G is 2-connected,consider v1, . . . , vn an st-ordering of G :

v3 v2

v1 v4 v5 v6

Ai = {v : d+(v ) = i} and ai = |Ai| for i = 1, . . . , k − 1

Then, ck(G ) ≤Qn_i=1−1d+(vi)! ≤ k!Qk_i=1−1(i!)ai

under Pk−1

a = n − 2 and Pk−1

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Number of k-edge colorings of k-regular graphs

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There exists a branching algorithm with running time O⋆((k − 1!)n2) and

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Enum-k-edge coloring:

There exists a branching algorithm with running time O⋆((k − 1!)n2) and

polynomial space to solve Enum-k-edge coloring. Proof: d+_(v 1)! = k! d+_(v 2)! At most Qn−1 i=1 d+(vi)! = O⋆((k − 1!) n 2) leaves

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Corollary

Let G be a cubic graph, then c3(G ) ≤ 3 · 2

n 2.

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Number of 3-edge colorings for simple graphs

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Can we improve a lot? Not really: let Hnbe:

Lemma (c3 of the ladder graph)

c (H ) =

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Number of 3-edge colorings for simple graphs

c3(Hn) =

2n/2+ 8, if n/2 is even,

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c (H ) =

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Number of 3-edge colorings for simple graphs

c3(Hn) =

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c (H ) =

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Number of 3-edge colorings for simple graphs

c3(Hn) =

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c (H ) =

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Number of 3-edge colorings for simple graphs

Can we improve a lot? Not really: let Mn be:

Lemma (c3 of the Mœbius ladder graph)

c3(Mn) =

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So, for simple cubic graphs, we have:

2n2 < max{c₃(G ) : G cubic graph on n vertices } ≤ 3.2

n 2

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Number of 3-edge colorings for simple graphs

2n2 < max{c₃(G ) : G cubic graph on n vertices } ≤

3.2n2 9 4.2 n 2 Theorem

Let G be a connected simple cubic graph. Then c3(G ) ≤ 9₄ · 2n−

n3 2.

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2n2 < max{c₃(G ) : G cubic graph on n vertices } ≤

3.2n2 9 4.2 n 2 Theorem

Let G be a connected simple cubic graph. Then c3(G ) ≤ 9₄ · 2n−

n3 2.

Proof:

vi

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4-total colorings

Let cT

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Let cT

4 (G ) be the number of 4-total colorings of G .

Theorem

Let G be a (2-)connected cubic graph. Then cT

4 (G ) ≤ 3 · 2

3n 2.

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4-total colorings

Let cT

4 (G ) be the number of 4-total colorings of G .

Theorem

4 (G ) ≤ 3 · 2 3n 2. Proof: v3 v2 v1 v4 v5 v6

A+= {v : d+(v ) = 2 and d−(v ) = 1}

A−= {v : d+(v ) = 1 and d−(v ) = 2}

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Theorem

4 (G ) ≤ 3 · 2

3n 2.

Proof: We have |A+| = |A−| = (n − 2)/2

Denote by ci the number of partial 4-total colorings of vertices and arcs

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4-total colorings

Theorem

4 (G ) ≤ 3 · 2

3n 2.

Proof: We have |A+| = |A−| = (n − 2)/2

with tail in {v1, . . . , vi}.

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Theorem

4 (G ) ≤ 3 · 2

3n 2.

Proof: We have |A+| = |A−| = (n − 2)/2

with tail in {v1, . . . , vi}.

c1 = 24

if vi ∈ A−

vi

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4-total colorings

Theorem

4 (G ) ≤ 3 · 2

3n 2.

Proof: We have |A+| = |A−| = (n − 2)/2

with tail in {v1, . . . , vi}. c1 = 24 if vi ∈ A− vi ci ≤ 2ci−1 if vi ∈ A+ vi ci ≤ 4ci−1

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Theorem

4 (G ) ≤ 3 · 2

3n 2.

Proof: We have |A+| = |A−| = (n − 2)/2

with tail in {v1, . . . , vi}. c1 = 24 if vi ∈ A− vi ci ≤ 2ci−1 if vi ∈ A+ vi ci ≤ 4ci−1 Hence: cT 4 (G ) ≤ 24 · 2 (n−2) 2 · 4 (n−2) 2 = 3 · 23n2

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Enum-4-total coloring:

There exists a branching algorithm with running time O⋆(23n2) =

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There exists a branching algorithm with running time O⋆(23n2) =

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Enum-4-total coloring:

A lower bound on cT

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4 , the nb of 4-total colorings:

Lemma

Let T be a binary tree (degree = 1 or 3) on n vertices, cT

4 (T ) = 3 · 2

3n 2.

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Enum-4-total coloring:

Lemma

4 (T ) = 3 · 2

3n 2.

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Lemma

4 (T ) = 3 · 2 3n 2. Proof: by induction: cT 4 (K1,3) = 4.3!.23 = 3.26 = 3.2 3.4 2

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Enum-4-total coloring:

Lemma

4 (T ) = 3 · 2 3n 2. Proof: by induction: cT 4 (K1,3) = 4.3!.23 = 3.26 = 3.2 3.4 2 cT 4 (T ) = 8.c4T(T \ {x, y }) x y z t

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Lemma

4 (T ) = 3 · 2 3n 2. Proof: by induction: cT 4 (K1,3) = 4.3!.23 = 3.26 = 3.2 3.4 2 cT 4 (T ) = 8.c4T(T \ {x, y }) x y z t

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Enum-4-total coloring:

Lemma

Let Tn be a binary tree on n vertices + parallel edges between twin leaves,

cT 4 (Tn) = √3₂· 2 5n 4. Proof: x y z t n = p − 2 + p + 2p

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Lemma

cT

4 (Tn) = √3₂· 2

5n 4.

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Enum-4-total coloring:

Lemma

cT 4 (Tn) = √3₂· 2 5n 4. Proof: cT 4 (T′) = 3 · 2 3(2p−2) 2

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Lemma

cT 4 (Tn) = √3₂· 2 5n 4. Proof: x y z t

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Enum-4-total coloring:

Lemma

cT 4 (Tn) = √3₂· 2 5n 4. Proof: x y z t cT 4 (T′) = 3 · 2 3(2p−2) 2 · 4p = 3 · 25p−3= 3 · 25n4−12

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k-edge colorings

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Some (hard?) open questions

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Some (hard?) open questions

Is there a O⋆((2 − ǫ)n_{) algorithm for some ǫ > 0}

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Minimum edge coloring in O⋆(2m₎

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Some (hard?) open questions

Is there a O⋆(2n_{) algorithm}

3-vertex coloring in O⋆_(1.3289n_{) (R. Beigel and D. Eppstein,}

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2005)

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Some (hard?) open questions

2005)

Is it possible to do better?

Under ETH, there is no O⋆₍₂o(n)_{) algorithm for 3-vertex coloring}

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Some (less hard?) open questions

(multi-graph).

Is it possible to fix

c3(n) = max{c3(G ) : G simple graph on n vertices}

We know that 2n2 < c₃(n) ≤ 9

4 · 2

n 2

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(multi-graph).

4 · 2

n 2

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Some (less hard?) open questions

(multi-graph).

4 · 2

n 2

Is it possible to find the algorithm with best running time for Enum-4-total coloring.

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