# L(p, q)-Labeling and Integer Flow on Planar Graphs

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A total k-coloring of a graph G is a coloring of V ( G) âˆª E( G) using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number Ï‡â€³( G) is the smallest integer k such that G has a total k-coloring. It is known that if a planar graph...

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A subset of a finite set of points in the plane is called an empty convex polygon or a hole if it forms the set of vertices of a convex polygon whose interior contains no points of the set. Let n(k, l, s) be the smallest integer such that any set of n(k, l, s) points in the plane, no three...

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In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity...

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Let N(k, l) be the smallest positive integer such that any set of N(k, l) points in general position in the plane contains a disjoint pair of a convex k-gon and a convex l-gon. In this paper, we mainly prove that 17 â‰¤ N(4, 6) â‰¤ 21.

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We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Î£ and integers Î” and k, determine the maximum order N(Î”, k,Î£) of a graph embeddable in Î£ with maximum degree Î” and diameter k. We introduce a number of constructions which...

- Edge coloring by total labelings of outerplanar graphs. Wang, Guang Hui; Yan, Gui Ying // Acta Mathematica Sinica;Nov2013, Vol. 29 Issue 11, p2129
An edge coloring total k-labeling is a labeling of the vertices and the edges of a graph G with labels {1, 2, â€¦, k} such that the weights of the edges define a proper edge coloring of G. Here the weight of an edge is the sum of its label and the labels of its two end vertices. This concept...