Corpus ID: 44014247

The chromatic number of the square of subcubic planar graphs

@article{Hartke2016TheCN,
  title={The chromatic number of the square of subcubic planar graphs},
  author={Stephen G. Hartke and Sogol Jahanbekam and Brent J. Thomas},
  journal={arXiv: Combinatorics},
  year={2016}
}
Wegner conjectured in 1977 that the square of every planar graph with maximum degree at most $3$ is $7$-colorable. We prove this conjecture using the discharging method and computational techniques to verify reducible configurations. 
The square of a planar cubic graph is 7-colorable
  • C. Thomassen
  • Mathematics, Computer Science
  • J. Comb. Theory, Ser. B
  • 2018
TLDR
It is proved that the conjecture made by G.Wegner in 1977 that the square of every planar, cubic graph is $7$-colorable is proved. Expand
Bounding the chromatic number of squares of K4-minor-free graphs
TLDR
It is proved that if G contains no subgraph isomorphic to K 2, r for some r ≥ 1, then χ ( G 2 ) ≤ Δ ( G ) + r . Expand
2-distance 4-coloring of Planar Subcubic Graphs with Girth at Least 21
TLDR
It is proved the existence of a 2-distance 4-coloring for planar subcubic graphs with girth at least 21 and a construction of a planarSubcUBic graph of girth 11 that is not 2- distance 4-colorable is shown. Expand
Counterexamples to Thomassen’s Conjecture on Decomposition of Cubic Graphs
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blueExpand
2-distance list (Δ+ 3)-coloring of sparse graphs
  • Hoang La
  • Computer Science, Mathematics
  • ArXiv
  • 2021
TLDR
The existence of a 2-distance list (∆ + 3)-coloring for graphs with maximum average degree less than 8 3 and maximum degree ∆ ≥ 4 is proved. Expand
An introduction to the discharging method via graph coloring
TLDR
The aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples. Expand
r-hued coloring of planar graphs with girth at least 8
In this paper, we consider simple undirected graphs without loops. A proper k-coloring of the vertices of a graph G = (V,E) is an assignment of colors from 1 to k such that no two adjacent verticesExpand
Distance-Two Colorings of Barnette Graphs
TLDR
It is claimed that the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which are called type-one Barnette graphs, since they are the first class identified by Barnette, and the problem is polynomial for cubic plane graphs with face sizes $3, 4, 5, $ or $6$, which are call type-two Barnettes graphs, because of their relation to Barnette's second conjecture. Expand
$2$-distance $(\Delta+2)$-coloring of sparse graphs
TLDR
It is proved the existence of a 2-distance (∆ + 2)-coloring for graphs with maximum average degree less than 8 3 and maximum degree ∆ ≥ 6 and as a corollary, every planar graph with girth at least 8 and maximum level of degree ≥ 6 admits a two-distance k-coloring. Expand
hued ( r + 1 )-coloring of planar graphswith irth at least 8 for r ≥ 9
Let r, k ≥ 1 be two integers. An r-hued k-coloring of the vertices of a graph G = (V , E) is a proper k-coloring of the vertices, such that, for every vertex v ∈ V , the number of colors in itsExpand
...
1
2
...

References

SHOWING 1-10 OF 23 REFERENCES
List Colouring Squares of Planar Graphs
TLDR
It is shown that the chromatic number of the square of every planar graph G with maximum degree Δ ⩾ 8 is at most ⌈ 3 2 Δ ⌉ + 1, and indeed this is true for the list Chromatic number. Expand
Choosability of the square of planar subcubic graphs with large girth
  • F. Havet
  • Computer Science, Mathematics
  • Discret. Math.
  • 2009
We show that the choice number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choice number of the square of a subcubicExpand
Precise Upper Bound for the Strong Edge Chromatic Number of Sparse Planar Graphs
Abstract We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40+1. The bound 2∆−1 is reached at any graph that has two adjacent vertices ofExpand
Choosability of the square of a planar graph with maximum degree four
TLDR
New upper bounds for the square of a planar graph with maximum degree $\Delta \leq 4$ are presented, in particular $G^2$ is 5-, 6-, 7-, 8-, 12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3 respectively. Expand
2-distance 4-colorability of Planar Subcubic Graphs with Girth at Least 22
TLDR
The 2-distance 4-colorability of planar subcubic graphs with g ≥ 22 is proved, and it is proved that there are graphs with arbitrarily large ∆ and g ≤ 6 having χ2(G) ≥ ∆ + 2. Expand
The Four-Colour Theorem
TLDR
Another proof is given, still using a computer, but simpler than Appel and Haken's in several respects, that every loopless planar graph admits a vertex-colouring with at most four different colours. Expand
THE STRONG CHROMATIC INDEX OF GRAPHS
Problems and results are presented concerning the strong chromatic index, where the strong chromatic index is the smallest k such that the edges of the graph can be k-colored with the property thatExpand
A bound on the chromatic number of the square of a planar graph
TLDR
The bound of the chromatic number of the square of any planar graph G with maximum degree Δ ≥ 8 is bounded by χ(G2) ≤ ⌊3/2 Δ⌋ + 1, and this is asymptotically an improvement on the previously best-known bound. Expand
An unavoidable set of D-reducible configurations
We give a new proof of the four-color theorem by exhibiting an unavoidable set of 2822 D-reducible configurations. The existence of such a set had been conjectured by several researchers includingExpand
On strong edge-colouring of subcubic graphs
TLDR
It is shown that every subcubic graph with maximum average degree strictly less than 73 can be strongly edge-coloured with six colours, and it is proved thatevery subcUBic planar graph without 4-cycles and 5-cycles can be strong edge-Coloured with nine colours. Expand
...
1
2
3
...