• 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 the Korean Mathematical Society
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• v.42 no.4
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• pp.847-856
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• 2005

M. Junger, G. Reinelt, and W. R. Pulleyblank asked the following questions ([2]). (1) Is it true that every simple planar 2-edge connected bipartite graph has a 3-partition in which each component consists of the edge set of a simple path? (2) Does every simple planar 2-edge connected graph have a 3-partition in which every component consists of the edge set of simple paths and triangles? The purpose of this paper is to provide a positive answer to the second question for simple outerplanar 2-vertex connected graphs and a positive answer to the first question for simple planar 2-edge connected bipartite graphs one set of whose bipartition has at most 4 vertices.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.263-271
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• 2024

A vertex coloring of a graph G is called injective if any two vertices with a common neighbor receive distinct colors. A graph G is injectively k-choosable if any list L of admissible colors on V (G) of size k allows an injective coloring 𝜑 such that 𝜑(v) ∈ L(v) whenever v ∈ V (G). The least k for which G is injectively k-choosable is denoted by χ^l_i(G). For a planar graph G, Bu et al. proved that χ^l_i(G) ≤ ∆ + 6 if girth g ≥ 5 and maximum degree ∆(G) ≥ 8. In this paper, we improve this result by showing that χ^l_i(G) ≤ ∆ + 6 for g ≥ 5 and arbitrary ∆(G).

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.139-151
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• 2016

A k-total-coloring of a graph G is a coloring of $V{\cup}E$ using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number ${\chi}^{{\prime}{\prime}}(G)$ of G is the smallest integer k such that G has a k-total-coloring. Let G be a planar graph with maximum degree ${\Delta}$. In this paper, it's proved that if ${\Delta}{\geq}7$ and G does not contain adjacent 5-cycles, then the total chromatic number ${\chi}^{{\prime}{\prime}}(G)$ is ${\Delta}+1$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.145-154
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• 2012

Let G be a planar graph with maximum degree $\Delta$. The linear 2-arboricity $la_2$(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. In this paper, we prove that (1) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+8$ if G has no adjacent 3-cycles; (2) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+10$ if G has no adjacent 4-cycles; (3) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+6$ if any 3-cycle is not adjacent to a 4-cycle of G.

• 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.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.391-402
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• 2022

For a given graph G = (V, E), a dominating set is a subset V' of the vertex set V so that each vertex in V ＼ V' is adjacent to a vertex in V'. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G). For an integer k ≥ 1, the k-th power G^k of a graph G with V (G^k) = V (G) for which uv ∈ E(G^k) if and only if 1 ≤ d_G(u, v) ≤ k. Note that G² is the square graph of a graph G. In this paper, we obtain some tight bounds for the sum of the domination numbers of a graph and its square graph in terms of the order, order and size, and maximum degree of the graph G. Also, we characterize such extremal graphs.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.331-342
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• 2017

Let R be a commutative ring with nonzero identity and G be a nontrivial finite group. Also, let Z(R) be the set of zero-divisors of R and, for $a{\in}Z(R)$, let $ann(a)=\{r{\in}R{\mid}ra=0\}$. The annihilator graph of the group ring RG is defined as the graph AG(RG), whose vertex set consists of the set of nonzero zero-divisors, and two distinct vertices x and y are adjacent if and only if $ann(xy){\neq}ann(x){\cup}ann(y)$. In this paper, we study the annihilator graph associated to a group ring RG.

• Journal of the Korea Society of Computer and Information
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• v.19 no.5
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• pp.19-25
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• 2014

In this paper, I seek the chromatic number, the maximum number of colors necessary when adjoining vertices in the plane separated apart at the distance of 1 shall receive distinct colors. The upper limit of the chromatic number has been widely accepted as $4{\leq}{\chi}(G){\leq}7$ to which Hadwiger-Nelson proposed ${\chi}(G){\leq}7$ and Soifer ${\chi}(G){\leq}9$ I firstly propose an algorithm that obtains the minimum necessary chromatic number and show that ${\chi}(G)=3$ is attainable by determining the chromatic number for Hadwiger-Nelson's hexagonal graph. The proposed algorithm obtains a chromatic number of ${\chi}(G)=4$ assuming a Hadwiger-Nelson's hexagonal graph of 12 adjoining vertices, and again ${\chi}(G)=4$ for Soifer's square graph of 8 adjoining vertices. assert. Based on the results as such that this algorithm suggests the maximum chromatic number of a planar graph is ${\chi}(G)=4$ using simple assigned rule of polynomial time complexity to color for a vertex with minimum degree.