The Crossing Numbers of Join of Paths and Cycles with Two Graphs of Order Five

Mari,; Kle, n; Schrötter, Štefan

doi:10.1007/978-3-642-28212-6_15

Cited by 32 publications

(46 citation statements)

References 5 publications

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“…It is also important to note that the crossing numbers of the graphs G + D n are known for few graphs G of order five and six, see e.g. [6,8,[10][11][12][13][14]. In all these cases, the graph G is usually connected and contains at least one cycle.…”

Section: Michal Stašmentioning

confidence: 99%

“…In [2,12,13], the properties of cyclic permutations are verified with the help of the software described in [1]. In our opinion, the methods used in [6,8,9] do not suffice for establishing the crossing number of the join product G * + D n . Some parts of proofs can be done with the help of software that generates all cyclic permutations in [1].…”

Section: Michal Stašmentioning

confidence: 99%

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On the crossing numbers of join products of five graphs of order six with the discrete graph

Staš¹

2020

Opuscula Math.

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The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G * + Dn, where the disconnected graph G * of order six consists of one isolated vertex and of one edge joining two nonadjacent vertices of the 5-cycle. In our proof, the idea of cyclic permutations and their combinatorial properties will be used. Finally, by adding new edges to the graph G * , the crossing numbers of Gi + Dn for four other graphs Gi of order six will be also established.

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Section: Michal Stašmentioning

confidence: 99%

Section: Michal Stašmentioning

confidence: 99%

On the crossing numbers of join products of five graphs of order six with the discrete graph

Staš¹

2020

Opuscula Math.

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“…Recall that the join product (or shortly, join) G + H of two graphs G and H is obtained from vertex-disjoint copies of G and H by adding all edges between V (G) and V (H). In [15], Kulli and Muddebihal characterized planar joins; the crossing numbers of joins of special graphs were studied in [10][11][12][13] and [16] (the crossing number cr(G) of a graph G is the minimum number of crossings in any plane drawing of G).…”

Section: Introductionmentioning

confidence: 99%

Joins of 1-planar graphs

Czap

Hudak

Madaras

2014

Acta. Math. Sin.-English Ser.

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A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join $G+H$ is 1-planar if and only if the pair $[G,H]$ is subgraph-majorized (that is, both $G$ and $H$ are subgraphs of graphs of the major pair) by one of pairs $[C_3 \cup C_3,C_3], [C_4,C_4], [C_4,C_3], [K_{2,1,1},P_3]$ in the case when both factors of the graph join have at least three vertices. If one factor has at most two vertices, then we give several necessary/sufficient conditions for the bigger factor

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“…Moreover, the exact values for crossing numbers of G + D n and G + P n for all graphs G of order at most four are given in [3]. Furthermore, the crossing numbers of the graphs G + D n are known for a few graphs G of order five and six in [4][5][6][7][8][9][10]. In all of these cases, the graph G is connected and contains at least one cycle.…”

Section: Introductionmentioning

confidence: 99%

Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

Staš

2019

Symmetry

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The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph D n and with the path P n on n vertices for three other graphs.

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The Crossing Numbers of Join of Paths and Cycles with Two Graphs of Order Five

Cited by 32 publications

References 5 publications

On the crossing numbers of join products of five graphs of order six with the discrete graph

On the crossing numbers of join products of five graphs of order six with the discrete graph

Joins of 1-planar graphs

Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

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