• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.577-587
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• 2006

We present several properties concerning infinite maximal planar graphs. Results related to the infinite VAP-free planar graphs are also included. Finally, we extend the result of W. Goddard, who showed that every finite 4-connected maximal planar graph is 4-ordered, to infinite strong triangulations.

• Journal of applied mathematics & informatics
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• v.27 no.3_4
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• pp.531-539
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• 2009

In this paper we first present a structure theorem for an infinite locally finite 3-connected VAP-free planar graph, and in connection with this result we study a possible classification of infinite locally finite planar graphs by reducing modulo finiteness.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.643-651
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• 2005

In the first part of this paper we investigate several statements concerning infinite maximal planar graphs which are equivalent in finite case. In the second one, for a given induced $\theta$-path (a finite induced path whose endvertices are adjacent to a vertex of infinite degree) in a 4-connected VAP-free maximal planar graph containing a vertex of infinite degree, a new $\theta$-path is constructed such that the resulting fan is tight.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.537-543
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• 2005

A graph is k-cyclable if given k vertices there is a cycle that contains the k vertices. Sallee showed that every finite 3-connected planar graph is 5-cyclable. In this paper Sallee's result is extended to 3-connected infinite locally finite VAP-free plane graphs containing no unbounded faces.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.83-93
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• 2006

A graph is k-cyclable if given k vertices there is a cycle that contains the k vertices. Sallee showed that every finite 3-connected planar graph is 5-cyclable. In this paper, by characterizing the circuit graphs and investigating the structure of LV-graphs, we extend his result to 3-connected infinite locally finite VAP-free plane graphs.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.1-21
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• 1996

In this paper the existence of spanning 3-trees in every 3-connected locally finite vertex-accumulation-point-free planer graph is verified, which is an extension of D. Barnette to infinite graphs and which improves the result of the author.

• 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.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.27-40
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• 2002

For two integers m, n with m $\leq$ n, an [m,n]-factor F in a graph G is a spanning subgraph of G with m $\leq$ d$\_$F/(v) $\leq$ n for all v ∈ V(F). In 1996, H. Enomoto et al. proved that every 3-connected Planar graph G with d$\_$G/(v) $\geq$ 4 for all v ∈ V(G) contains a [2,3]-factor. In this paper. we extend their result to all 3-connected locally finite infinite planar graphs containing no unbounded faces.

• 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.

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.249-255
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• 2021

The aim of this work is to study the discreteness of the spectrum of the Schrödinger operator on infinite quantum graphs in a magnetic field. The problem was solved on a set of quantum graphs of a special kind.