Bounds on the number of tessellations in graphs Alexandre S. Abreu 1 - - PowerPoint PPT Presentation

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Bounds on the number of tessellations in graphs Alexandre S. Abreu 1 - - PowerPoint PPT Presentation

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Tessellations Results Bounds on the number of tessellations in graphs Alexandre S. Abreu 1 , Lu s Felipe I. Cunha 1 Franklin L. Marquezino 1 and Luis Antonio B. Kowada 2 1 PESC/COPPE-UFRJ 2 IC-UFF November 11, 2016 A.S. Abreu et al.

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Tessellations Results

Bounds on the number of tessellations in graphs

Alexandre S. Abreu1, Lu´ ıs Felipe I. Cunha1 Franklin L. Marquezino1 and Luis Antonio B. Kowada2

1 PESC/COPPE-UFRJ 2 IC-UFF

November 11, 2016

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Outline

1 Tessellations in Graphs 2 Bounds on the number of tessellations

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Section 1 Tessellations in Graphs

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Definition

Let Ti = {c1, c2, · · · , cn} be a family of cliques of a graph G; Ti is a tessellation in G if and only if:

All cliques of Ti are disjoint, and; The union of the cliques of Ti covers all vertices of G.

Each clique in Ti is called a cluster (Portugal, 16).

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Example

Figure 1: Tessellations of G. Each tessellation covers all vertices.

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Number of Tessellations of a Graph

A graph G is T-tessellable if T is the smallest number of tessellations such that the union of these tessellations covers all edges of G.

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Number of Tessellations of a Graph

A graph G is T-tessellable if T is the smallest number of tessellations such that the union of these tessellations covers all edges of G.

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Number of Tessellations of a Graph

A graph G is T-tessellable if T is the smallest number of tessellations such that the union of these tessellations covers all edges of G.

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Number of Tessellations of a Graph

A graph G is T-tessellable if T is the smallest number of tessellations such that the union of these tessellations covers all edges of G.

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Motivation: Quantum walks

Quantum walks are the model of a particle’s tour through the vertices of a graph. In quantum walks there are quantum state representing the walker, and an evolution operator applied on the quantum state, moving the walker through the graph’s vertices. Staggered quantum walks model uses tessellations in graphs to generate the evolution operators (Portugal et al., 16)

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2-tessellable graphs

Proposition 1 ((PORTUGAL, 2016)) A graph is 2-tessellable if and only if its clique graph is 2-colorable. This proposition is not generalizable for T > 2.

The characterization of a T-tessellable graph is still an open problem.

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Section 2 Bounds on the number of tessellations

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Upper Bound

We propose an upper bound for the number of tessellations. Lemma 1 Given G and its clique graph K(G), then T(G) ≤ χ(K(G)).

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The proof’s Idea of Lemma 1

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The proof’s Idea of Lemma 1

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The proof’s Idea of Lemma 1

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The proof’s Idea of Lemma 1

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Upper bound is tight

Theorem 1 Let G be a graph s.t. there is a vertex v ∈ V (G) which is a cut vertex and all cliques of G just share v. Then, T(G) = χ(K(G)).

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The proof’s Idea of Theorem 1

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The proof’s Idea of Theorem 1

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The proof’s Idea of Theorem 1

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The proof’s Idea of Theorem 1

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The proof’s Idea of Theorem 1

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The proof’s Idea of Theorem 1

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Lower Bound

We propose a lower bound for the number of tessellations. Lemma 2 Let G be a graph and K(G) its clique graph. Then, T(G) ≥ ⌈ χ(K(G))

2

⌉.

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The proof’s Idea of Lemma 2

1 k-chromatic graph has at least k vertex with degree at least

k − 1.

2 For every graph G, T(G) ≤ χ(K(G)) ≤ ∆(K(G)) + 1.

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The proof’s Idea of Lemma 2

Idea:

m = m(K(G)) =

  • i di

2

By (1) we have that m = χ(χ−1)

2

+ σ

2 , where σ is the

remaining sum, and 0 ≤ σ. T(G) ≤ χ(χ−1)

2

+ σ

2 .

So, χ2

2 − χ 2 + σ 2 − T(G) ≥ 0.

Let us divide our problem in three cases, considering φ ≥ 1:

(i)T(G) > χ

2 , i.e., T(G) = χ 2 + φ;

(ii)T(G) = χ

2 , and;

(iii)T(G) < χ

2 , i.e., T(G) = χ 2 − φ A.S. Abreu et al. November 11, 2016 16 / 21

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The proof’s Idea of Lemma 2

In each of this three cases, we have to solve an inequation. By this inequation, it comes that to get an answer in R the discriminant ∆ must be greater than or equal to zero. So:

(i)∆ = 1 − σ + 2φ ≥ 0 → σ ≤ 1 + 2φ; (ii)∆ = 1 − σ ≥ 0 → σ ≤ 1, however; (iii)∆ = 1 − σ − 2φ ≥ 0 → σ ≤ 1 − 2φ < 0, but σ ≥ 0.

Thus, we never have that T(G) < χ(K(G))

2

.

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Lower bound is tight

For a wheel graph w-wheel, s.t. w > 4, we have that T(G) = ⌈ χ

2 ⌉.

Theorem 2 Let G be a w-wheel, s.t. w > 4. Then, T(G) = ⌈ χ

2 ⌉.

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Tight Bounds

Theorem 3 Let G be a graph and K(G) its clique graph, and let K(G) is non-bipartite. We have that ⌈ χ(K(G))

2

⌉ ≤ T(G) ≤ χ(K(G)).

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Open Questions

What is the number of tessellations for other classes? What is the complexity of this problem?

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References

PORTUGAL, R. Staggered quantum walks on graphs. arXiv preprint arXiv:1603.02210, 2016. PORTUGAL, R. et al. The staggered quantum walk model. Quantum Information Processing, Springer, v. 15, n. 1, p. 85–101, 2016.

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