Combinatorial Geometry and Graph Theory: Indonesia-Japan by Jin Akiyama, Edy Tri Baskoro, Mikio Kano

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By Jin Akiyama, Edy Tri Baskoro, Mikio Kano

This publication constitutes the completely refereed post-proceedings of the Indonesia-Japan Joint convention on Combinatorial Geometry and Graph conception, IJCCGGT 2003, held in Bandung, Indonesia in September 2003.

The 23 revised papers provided have been conscientiously chosen in the course of rounds of reviewing and development. one of the issues coated are coverings, convex polygons, convex polyhedra, matchings, graph colourings, crossing numbers, subdivision numbers, combinatorial optimization, combinatorics, spanning bushes, a number of graph characteristica, convex our bodies, labelling, Ramsey quantity estimation, etc.

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Extra info for Combinatorial Geometry and Graph Theory: Indonesia-Japan Joint Conference, IJCCGGT 2003, Bandung, Indonesia, September 13-16, 2003, Revised Selected Papers

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A7 ]. Then {H ∗ |H ∈ D8 } is a uniform covering of 2-paths with 6-paths in K7 . (2) n = 10 For a Hamilton cycle in D10 , H = (0, a1 , a2 , . . , a9 ), where a1 < a9 , define two 6-paths H ∗ , H ∗∗ : H ∗ = [0, a1 , a2 , a3 , a4 , a5 , a6 ], H ∗∗ = [a5 , a6 , a7 , a8 , a9 , 0, a1 ]. Then {H ∗ , H ∗∗ |H ∈ D10 } is a uniform covering of 2-paths with 6-paths in K10 . (3) n = 11 Define σ11 : σ11 = (0 1 2 · · · 10) so that σ11 is a vertex-rotation in K11 . Put P1 = [2, 8, 3, 7, 4, 6, 5], P2 = [10, 1, 8, 3, 6, 5, 4], P3 = [4, 7, 1, 10, 9, 2, 6], P4 = [9, 2, 5, 6, 1, 10, 8], P5 = [10, 7, 4, 2, 9, 8, 3], P6 = [0, 10, 1, 9, 2, 7, 4], P7 = [0, 9, 2, 1, i 10, 4, 7], P8 = [1, 10, 5, 6, 9, 2, 8], P9 = [8, 3, 2, 9, 7, 4, 5].

Since m ≥ 11, we have r ≥ 5. Define r 6-paths as follows: R= k [d, −(k + 1), k, a, −k, k + 1, e] R= r [e, −(r − 1), −r, a, r, r − 1, d]. (1 ≤ k ≤ r − 1), Put R = T {Rk | 1 ≤ k ≤ r}. 2 When m is odd, π(R) ⊃ {[x, a, y], [a, x, y] | x, y ∈ Vm , x = y }. Proof. For any l (1 ≤ l ≤ r), a 2-path [x, a, y] with d(x, y) = l belongs to R. Therefore π(R) ⊃ {[x, a, y] | x, y ∈ Vm , x = y }. For any l (1 ≤ l ≤ r), both a 2-path [a, x, y] with y − x = l and a 2-path [a, x, y] with y − x = −l belong to R. Therefore π(R) ⊃ {[a, x, y] | x, y ∈ Vm , x = y}.

5. M. Kobayashi and G. Nakamura, Uniform coverings of 2-paths by 4-paths, Australasian J. Combin. 24 (2001) 301-304. 6. M. Kobayashi, G. Nakamura and C. Nara, Uniform coverings of 2-paths with 5-paths in K2n , Australasian J. Combin. 27 (2003) 247-252. 7. M. Kobayashi, G. Nakamura and C. Nara, Uniform coverings of 2-paths with 5-paths in the complete graph, accepted. 8. M. Kobayashi and G. Nakamura, Uniform coverings of 2-paths with 6-cycles in the complete graph, manuscript. jp Abstract. e. the n-gon is a net of the polyhedron.

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