On square-free vertex colorings of graphs

2007 ◽  
Vol 44 (3) ◽  
pp. 411-422 ◽  
Author(s):  
János Barát ◽  
Péter Varjú
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Path Following ◽  
Fixed Number ◽  
The Other ◽  
Graph Colorings ◽  
Outerplanar Graphs ◽  
Free Vertex ◽  
Other Hand ◽  
Color Sequence

A sequence of symbols a1 , a2 … is called square-free if it does not contain a subsequence of consecutive terms of the form x1 , …, xm , x1 , …, xm . A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Hałuszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k -tree have a coloring of this kind using O ( ck ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

scholarly journals Dot Product Representations of Planar Graphs

10.37236/703 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ross J. Kang ◽  
László Lovász ◽  
Tobias Müller ◽  
Edward R. Scheinerman
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Discrete Math ◽  
The Other ◽  
Outerplanar Graphs ◽  
Product Graph ◽  
Other Hand ◽  
Vector Labelling ◽  
Dot Product

A graph $G$ is a $k$-dot product graph if there exists a vector labelling $u: V(G) \to \mathbb{R}^k$ such that $u(i)^{T}u(j) \geq 1$ if and only if $ij \in E(G)$. Fiduccia, Scheinerman, Trenk and Zito [Discrete Math., 1998] asked whether every planar graph is a $3$-dot product graph. We show that the answer is "no". On the other hand, every planar graph is a $4$-dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.

scholarly journals The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

scholarly journals The Size of Edge-critical Uniquely 3-Colorable Planar Graphs

10.37236/3228 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Naoki Matsumoto
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs ◽  
Infinite Family ◽  
The Other ◽  
Other Hand ◽  
Colorable Graph

A graph $G$ is uniquely $k$-colorable if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. In this paper, we prove that if $G$ is an edge-critical uniquely $3$-colorable planar graph, then $|E(G)|\leq \frac{8}{3}|V(G)|-\frac{17}{3}$. On the other hand, there exists an infinite family of edge-critical uniquely 3-colorable planar graphs with $n$ vertices and $\frac{9}{4}n-6$ edges. Our result gives a first non-trivial upper bound for $|E(G)|$.

On Cycles and Connectivity in Planar Graphs

1973 ◽  
Vol 16 (2) ◽  
pp. 283-288 ◽  
Author(s):  
M. D. Plummer ◽  
E. L. Wilson
Keyword(s):  
Planar Graphs ◽  
Hamiltonian Cycle ◽  
The Other ◽  
Other Hand

Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.

scholarly journals Triangulations and the Hajós Conjecture

10.37236/1982 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Bojan Mohar
Keyword(s):  
Projective Plane ◽  
Planar Graphs ◽  
The Other ◽  
Closed Surfaces ◽  
Other Hand

The Hajós Conjecture was disproved in 1979 by Catlin. Recently, Thomassen showed that there are many ways that Hajós conjecture can go wrong. On the other hand, he observed that locally planar graphs and triangulations of the projective plane and the torus satisfy Hajós Conjecture, and he conjectured that the same holds for arbitrary triangulations of closed surfaces. In this note we disprove the conjecture and show that there are different reasons why the Hajós Conjecture fails also for triangulations.

scholarly journals Further characterizations and Helly-property in k-trees

2016 ◽  
Vol 15 (3) ◽  
pp. 1-8
Author(s):  
H P Patil
Keyword(s):  
Planar Graphs ◽  
Similar Type ◽  
The Other ◽  
Forbidden Subgraphs ◽  
Outerplanar Graphs ◽  
Helly Property ◽  
Maximal Outerplanar Graphs

The purpose of this paper is to obtain a characterization of $k$-trees in terms of $k$-connectivity and forbidden subgraphs. Also, we present the other characterizations of $k$-trees containing the full vertices by using the join operation. Further, we establish the property of $k$-trees dealing with the degrees and formulate the Helly-property for a family of nontrivial $k$-paths in a $k$-tree. We study the planarity of $k$-trees and express the maximal outerplanar graphs in terms of 2-trees and $K_2$-neighbourhoods. Finally, the similar type of results for the maximal planar graphs are obtained.

SIMULTANEOUS EMBEDDING OF OUTERPLANAR GRAPHS, PATHS, AND CYCLES

2007 ◽  
Vol 17 (02) ◽  
pp. 139-160 ◽  
Author(s):  
EMILIO DI GIACOMO ◽  
GIUSEPPE LIOTTA
Keyword(s):  
Planar Graphs ◽  
The Other ◽  
Simple Path ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Simultaneous Embedding

Let G1 and G2 be two planar graphs having some vertices in common. A simultaneous embedding of G1 and G2 is a pair of crossing-free drawings of G1 and G2 such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle.

scholarly journals Non-Separating Planar Graphs

10.37236/8816 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hooman R. Dehkordi ◽  
Graham Farr
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Disjoint Paths ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Structural Characterisation ◽  
A Minor ◽  
Vertex Disjoint

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$. Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles. Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with $n$ vertices and $4n-10$ edges (the maximum possible) in 1983.

scholarly journals Path Choosability of Planar Graphs

10.37236/7139 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Glenn G. Chappell ◽  
Chris Hartman
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Vertex Coloring ◽  
List Coloring ◽  
Outerplanar Graphs ◽  
Color Class ◽  
Path Coloring

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

The Generalized Rank of Trace Languages

2019 ◽  
Vol 30 (01) ◽  
pp. 135-169
Author(s):  
Michal Kunc ◽  
Jan Meitner
Keyword(s):  
Infinite Sequence ◽  
Fixed Number ◽  
The Other ◽  
Direct Products ◽  
Commutative Monoids ◽  
Free Products ◽  
Free Monoids ◽  
Other Hand ◽  
Rational Series ◽  
Regular Sets

Given a partially commutative alphabet and a set of words [Formula: see text], the rank of [Formula: see text] expresses the amount of shuffling required to produce a word belonging to [Formula: see text] from two words whose concatenation belongs to the closure of [Formula: see text] with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.

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