By Jin Akiyama, Mikio Kano, Toshinori Sakai

This ebook constitutes the refereed complaints of the Thailand-Japan Joint convention on Computational Geometry and Graphs, TJJCCGG 2012, held in Bangkok, Thailand, in December 2012.

The 15 unique learn papers provided have been chosen from between six plenary talks, one specific public speak and forty-one talks by means of members from approximately 20 nations all over the world. TJJCCGG 2012 supplied a discussion board for researchers operating in computational geometry, graph theory/algorithms and their applications.

**Read or Download Computational Geometry and Graphs: Thailand-Japan Joint Conference, TJJCCGG 2012, Bangkok, Thailand, December 6-8, 2012, Revised Selected Papers PDF**

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**Additional resources for Computational Geometry and Graphs: Thailand-Japan Joint Conference, TJJCCGG 2012, Bangkok, Thailand, December 6-8, 2012, Revised Selected Papers**

**Sample text**

Thus the number of the a-colorings of La which contain Bi as a subset is aj=3 |Aj |. That is, the number of the a-colorings of La which contain Bi as a subset is at most 3a−2 . Note that in case a = 1, all a-colorings are 1-colorings; hence, the number of a-colorings containing Bi as a subset is 3a−2 = 0. Case 3. |Bi | = 3, say Bi = {r, s, t}. Consider an a-coloring of La containing r, s and t. Without loss of generality, suppose that r ∈ A1 , s ∈ A2 , t ∈ A3 . Again, we can choose the other a − 3 colors from each Aj where j = 4, 5, .

Let P ⊂ R2 be an n-point with |CH(P )| = m ≥ 4 even. Then any m quadrangulation of P has (n − 1) − m 2 quadrilaterals and 2(n − 1) − 2 edges. The following two lemmas are main tools for proving the main theorem: Lemma 3. Let Q ⊂ R2 be a properly m-sided simple convex polygon, where m ≥ 4 is even. If at least four color appears in Q or ω(Q) = 0, then Q can be properly colored quadrilaterals. partitioned into m−2 2 Proof. We proceed by induction on m. The case m = 4 is trivial, and thus we assume that the lemma holds for every m < m.

As a side remark, note that it is not clear that the property “there exists an n-universal point set of size n” is monotone in n. 2 Large Universal Point Sets A planar 3-tree is a maximal planar graph obtained by iteratively splitting a facial triangle into three new triangles with a degree-three vertex, starting from a single triangle. Since a planar 3-tree is a maximal planar graph, it has n vertices and 2n − 4 triangular faces and its combinatorial embedding is ﬁxed up to the choice of the outer face.