Triangle-Partitioning Edges of Planar Graphs, Toroidal Graphs and k-Planar Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

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Note on (semi-)proper orientation of some triangulated planar graphs

Applied Mathematics and Computation ◽

10.1016/j.amc.2020.125723 ◽

2021 ◽

Vol 392 ◽

pp. 125723

Author(s):

Ruijuan Gu ◽

Hui Lei ◽

Yulai Ma ◽

Zhenyu Taoqiu

Keyword(s):

Planar Graphs ◽

Proper Orientation

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Distributed Dominating Set Approximations beyond Planar Graphs

ACM Transactions on Algorithms ◽

10.1145/3326170 ◽

2019 ◽

Vol 15 (3) ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Saeed Akhoondian Amiri ◽

Stefan Schmid ◽

Sebastian Siebertz

Keyword(s):

Planar Graphs ◽

Dominating Set

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Clustered 3-colouring graphs of bounded degree

Combinatorics Probability Computing ◽

10.1017/s0963548321000213 ◽

2021 ◽

pp. 1-13

Author(s):

Vida Dujmović ◽

Louis Esperet ◽

Pat Morin ◽

Bartosz Walczak ◽

David R. Wood

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Bounded Degree ◽

Bounded Number ◽

Proper Vertex ◽

Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

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