scholarly journals Coloring with no $2$-Colored $P_4$'s

10.37236/1779 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Glenn G. Chappell ◽  
H. A. Kierstead ◽  
André Kündgen ◽  
Radhika Ramamurthi
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Short Proof ◽  
Cubic Graphs ◽  
Proper Coloring ◽  
Be Star ◽  
Tree Width ◽  
Star Coloring

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

scholarly journals Star Coloring High Girth Planar Graphs

10.37236/848 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Proper Coloring ◽  
Be Star ◽  
Star Coloring

A star coloring of a graph is a proper coloring such that no path on four vertices is 2-colored. We prove that every planar graph with girth at least 9 can be star colored using 5 colors, and that every planar graph with girth at least 14 can be star colored using 4 colors; the figure 4 is best possible. We give an example of a girth 7 planar graph that requires 5 colors to star color.

A Note on 3-choosability of Planar Graphs Related to Montanssier’s Conjecture

2016 ◽  
Vol 59 (2) ◽  
pp. 440-448
Author(s):  
Haihui Zhang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Proper Coloring

AbstractFor a given list assignment L = {L(v) : v ∈ V(G)}, a graph G = (V, E) is L-colorable if there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V. If G is L-colorable for every list assignment L having |L(v)| ≥ k for all v ∈ V, then G is said to be k-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.

scholarly journals Tree-width and large grid minors in planar graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Alexander Grigoriev
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Tree Decomposition ◽  
International Audience ◽  
Tree Width ◽  
Grid Minor ◽  
Large Grid

Graphs and Algorithms International audience We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.

scholarly journals KNOT-INEVITABLE PROJECTIONS OF PLANAR GRAPHS

1996 ◽  
Vol 05 (06) ◽  
pp. 877-883 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
TATSUYA TSUKAMOTO
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Spatial Graph ◽  
Torus Knot ◽  
Nonplanar Graph

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

scholarly journals Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

2010 ◽  
Vol 310 (21) ◽  
pp. 3049-3051 ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Short Proof

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

A Important Property on Planar Graph

2011 ◽  
Vol 50-51 ◽  
pp. 245-248
Author(s):  
Na Na Li ◽  
Shu Jun Wang ◽  
Xian Rui Meng
Keyword(s):  
Planar Graph ◽  
Planar Graphs

In 1994, C.Thomassen proved that every planar graph is (namely ). In 1993, M.Voigt shown that there are planar graphs which are not . But no one know whether every planar graph is . In this paper, we give a important property on planar graph that “Every planar graph is ” and “Every planar graph is free ” are equivalent.

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