• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.763-787
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• 2017

A graph is called 1-planar if it can be drawn in the Euclidean plane ${\mathbb{R}}^2$ such that each edge is crossed by at most one other edge. The weight of an edge is the sum of degrees of two ends. It is known that every planar graph of minimum degree ${\delta}{\geq}3$ has an edge with weight at most 13. In the present paper, we show the existence of edges with weight at most 25 in 3-connected 1-planar graphs.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.1
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• pp.7-11
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• 2008

The energy of a graph is the sum of the absolute values of its eigen values. We obtain some bounds for the energy of planar graphs in terms of its vertices, edges and faces.

• Journal of Korean Institute of Industrial Engineers
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• v.23 no.2
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• pp.359-370
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• 1997

We consider a facility layout problem with the objective of minimizing total transportation distance, which is the sum of rectilinear distances between facilities weighted by the frequency of trips between the facilities. It is assumed that facilities are required to have rectangular shapes and there is no empty space between the facilities in the layout. In this study, a graph theoretic heuristic is developed for the problem. In the heuristic, planar graphs are constructed to represent adjacencies between the facilities and then the graphs are converted to block layouts on a continual plane using a layout construction module. (Therefore, each graph corresponds to a layout.) An initial layout is obtained by constructing a maximal weighted planar graph and then the layout is improved by changing the planar graph. A simulated annealing algorithm is used to find a planar graph which gives the best layout. To show the performance of the proposed heuristic, computational experiments are done on randomly generated test problems and results are reported.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.105-115
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• 2003

An infinite locally finite plane graph is called an LV-graph if it is 3-connected and VAP-free. If an LV-graph G contains no unbounded faces, then we say that G is a 3LV-graph. In this paper, a structure theorem for an LV-graph concerning the existence of a sequence of systems of paths exhausting the whole graph is presented. Combining this theorem with the early result of the author, we obtain a necessary and sufficient conditions for an infinite VAP-free planar graph to be a 3LV-graph as well as an LV-graph. These theorems generalize the characterization theorem of Thomassen for infinite triangulations.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.27-43
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• 2022

The one-sided fattenings (called semi-ribbon graph in this paper) of the graph embedded in the real projective plane ℝℙ² are completely classified up to topological equivalence. A planar graph (i.e., embedded in the plane), admitting the one-sided fattening, is known to be a cactus boundary. For the graphs embedded in ℝℙ² admitting the one-sided fattening, unlike the planar graphs, a new building block appears: a bracelet along the Möbius band, which is not a connected summand of the oriented surfaces.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1305-1319
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• 2015

We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.359-365
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• 2012

Giving a planar graph G, let $x^'_l(G)$ and $x^{''}_l(G)$ denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if a planar graph G without 6-cycles with chord, then $x^'_l(G){\leq}{\Delta}(G)+1$ and $x^{''}_l(G){\leq}{\Delta}(G)+2$ where ${\Delta}(G){\geq}6$.

• Journal of KIISE:Computer Systems and Theory
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• v.33 no.7
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• pp.448-453
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• 2006

We extend a parallel depth-first search (DFS) algorithm for planar graphs to deal with (non-planar) solid grid graphs, a subclass of non-planar grid graphs. The proposed algorithm takes time O(log n) with $O(n/sqrt{log\;n})$ processors in Priority PRAM model. In our knowledge, this is the first deterministic NC algorithm for a non-planar graph class.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.511-517
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• 2014

In this paper, we prove that the 3-cycle is light in the family of 1-planar graphs with minimum vertex degree at least 5 and minimum edge degree at least 12. This generates a known result of Fabrici and Madaras [8].

• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.235-242
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• 1997

We characterize the tight structure of a vertex-accumula-tion-free maximal planar graph with no separating triangles. Together with the result of Halin who gave an equivalent form for such graphs this yields that a tight structure always exists in every 4-connected maximal planar graph with one end.