A new proof that 4-connected planar graphs are Hamiltonian-connected

Xiaoyun Lu; Douglas B. West

doi:10.7151/dmgt.1872

A new proof that 4-connected planar graphs are Hamiltonian-connected

Discussiones Mathematicae Graph Theory ◽

10.7151/dmgt.1872 ◽

2016 ◽

Vol 36 (3) ◽

pp. 555

Author(s):

Xiaoyun Lu ◽

Douglas B. West

Keyword(s):

Planar Graphs ◽

Hamiltonian Connected

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4-connected projective-planar graphs are Hamiltonian-connected

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2014.11.006 ◽

2015 ◽

Vol 112 ◽

pp. 36-69 ◽

Cited By ~ 2

Author(s):

Ken-ichi Kawarabayashi ◽

Kenta Ozeki

Keyword(s):

Planar Graphs ◽

Hamiltonian Connected

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4-connected projective-planar graphs are hamiltonian-connected

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms ◽

10.1137/1.9781611973105.28 ◽

2013 ◽

Cited By ~ 2

Author(s):

Ken-ichi Kawarabayashi ◽

Kenta Ozeki

Keyword(s):

Planar Graphs ◽

Hamiltonian Connected

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Acute Constraints in Straight-Line Drawings of Planar Graphs

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e102.a.994 ◽

2019 ◽

Vol E102.A (9) ◽

pp. 994-1001

Author(s):

Akane SETO ◽

Aleksandar SHURBEVSKI ◽

Hiroshi NAGAMOCHI ◽

Peter EADES

Keyword(s):

Planar Graphs ◽

Line Drawings ◽

Straight Line

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A Space-Efficient Separator Algorithm for Planar Graphs

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e102.a.1007 ◽

2019 ◽

Vol E102.A (9) ◽

pp. 1007-1016 ◽

Cited By ~ 1

Author(s):

Ryo ASHIDA ◽

Sebastian KUHNERT ◽

Osamu WATANABE

Keyword(s):

Planar Graphs

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Maximal distance spectral radius of 4-chromatic planar graphs

Linear Algebra and its Applications ◽

10.1016/j.laa.2021.02.005 ◽

2021 ◽

Vol 618 ◽

pp. 183-202

Author(s):

Aysel Erey

Keyword(s):

Spectral Radius ◽

Planar Graphs ◽

Maximal Distance

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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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