scholarly journals Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of GLn-Complexes

2001 ◽  
Vol 239 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Dikran B. Karaguezian ◽  
Victor Reiner ◽  
Michelle L. Wachs
Keyword(s):  
Bounded Degree ◽  
Graph Complexes ◽  
Degree Graph

scholarly journals Topology of bounded-degree graph complexes

2003 ◽  
Vol 262 (2) ◽  
pp. 287-312 ◽  
Author(s):  
Xun Dong
Keyword(s):  
Bounded Degree ◽  
Graph Complexes ◽  
Degree Graph

scholarly journals Computing the Number of Induced Copies of a Fixed Graph in a Bounded Degree Graph

Algorithmica ◽  
2018 ◽  
Vol 81 (5) ◽  
pp. 1844-1858 ◽  
Author(s):  
Viresh Patel ◽  
Guus Regts
Keyword(s):  
Bounded Degree ◽  
Degree Graph

Bounded degree graph inference from walks

COLT ◽  
1991 ◽  
pp. 354-366
Author(s):  
Vijay Raghavan
Keyword(s):  
Bounded Degree ◽  
Degree Graph ◽  
Graph Inference

scholarly journals KŐNIG’S LINE COLORING AND VIZING’S THEOREMS FOR GRAPHINGS

2016 ◽  
Vol 4 ◽  
Author(s):  
ENDRE CSÓKA ◽  
GÁBOR LIPPNER ◽  
OLEG PIKHURKO
Keyword(s):  
Maximum Degree ◽  
Edge Coloring ◽  
Classical Theorem ◽  
Orbit Equivalence ◽  
Bounded Degree ◽  
Finite Graphs ◽  
Edge Colorings ◽  
Graph Limits ◽  
Degree Graph ◽  
Odd Cycles

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.

scholarly journals Bounded degree graph inference from walks

1994 ◽  
Vol 49 (1) ◽  
pp. 108-132 ◽  
Author(s):  
Vijay Raghavan
Keyword(s):  
Bounded Degree ◽  
Degree Graph ◽  
Graph Inference

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

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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