Spanning Trees with Bounded Number of Branch Vertices

2002 ◽  
pp. 355-365 ◽  
Author(s):  
Luisa Gargano ◽  
Pavol Hell ◽  
Ladislav Stacho ◽  
Ugo Vaccaro
Keyword(s):  
Spanning Trees ◽  
Bounded Number

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

2012 ◽  
Vol 30 (2) ◽  
pp. 429-437 ◽  
Author(s):  
Haruhide Matsuda ◽  
Kenta Ozeki ◽  
Tomoki Yamashita
Keyword(s):  
Spanning Trees ◽  
Free Graph ◽  
Bounded Number

scholarly journals Spanning trees with a bounded number of leaves

2017 ◽  
Vol 37 (4) ◽  
pp. 501
Author(s):  
Junqing Cai ◽  
Evelyne Flandrin ◽  
Hao Li ◽  
Qiang Sun
Keyword(s):  
Spanning Trees ◽  
Bounded Number ◽  
Number Of Leaves

scholarly journals Clustered 3-colouring graphs of bounded degree

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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