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.

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.

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 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 $\frac{3n}{13}$.

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.

scholarly journals The maximum Wiener index of maximal planar graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1121-1135
Author(s):  
Debarun Ghosh ◽  
Ervin Győri ◽  
Addisu Paulos ◽  
Nika Salia ◽  
Oscar Zamora
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Wiener Index ◽  
Maximal Planar Graph

Abstract The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

scholarly journals THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 357-395 ◽  
Author(s):  
THERESE BIEDL ◽  
MARTIN VATSHELLE
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Line Drawing ◽  
Np Hard ◽  
Bounded Treewidth ◽  
Straight Line ◽  
Triangulated Graphs ◽  
Point Set ◽  
Fixed Embedding ◽  
Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

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