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 $|E(G)|$.

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.

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 Planar graphs have bounded nonrepetitive chromatic number

2020 ◽  
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Gwenaël Joret ◽  
Bartosz Walczak ◽  
David Wood
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs ◽  
Probabilistic Method ◽  
Color Pattern ◽  
Binary Representation ◽  
Strong Product ◽  
Minimum Number ◽  
Shaped Graph

The following seemingly simple question with surprisingly many connections to various problems in computer science and mathematics can be traced back to the beginning of the 20th century to the work of [Axel Thue](https://en.wikipedia.org/wiki/Axel_Thue): How many colors are needed to color the positive integers in a way such that no two consecutive segments of the same length have the same color pattern? Clearly, at least three colors are needed: if there was such a coloring with two colors, then any two consecutive integers would have different colors (otherwise, we would get two consecutive segments of length one with the same color pattern) and so the colors would have to alternate, i.e., any two consecutive segments of length two would have the same color pattern. Suprisingly, three colors suffice. The coloring can be constructed as follows. We first define a sequence of 0s and 1s recursively as follows: we start with 0 only and in each step we take the already constructed sequence, flip the 0s and 1s in it and append the resulting sequence at the end. In this way, we sequentially obtain the sequences 0, 01, 0110, 01101001, etc., which are all extensions of each other. The limiting infinite sequence is known as the [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence). Another view of the sequence is that the $i$-th element is the parity of the number of 1s in the binary representation of $i-1$, i.e., it is one if the number is odd and zero if it is even. The coloring of integers is obtained by coloring an integer $i$ by the difference of the $(i+1)$-th and $i$-th entries in the Thue-Morse sequence, i.e., the sequence of colors will be 1, 0, -1, 1, -1, 0, 1, 0, etc. One of the properties of the Thue-Morse sequence is that it does not containing two overlapping squares, i.e., there is no sequence X such that 0X0X0 or 1X1X1 would be a subsequence of the Thue-Morse sequence. This implies that the coloring of integers that we have constructed has no two consecutive segments with the same color pattern. The article deals with a generalization of this notion to graphs. The _nonrepetitive chromatic number_ of a graph $G$ is the minimum number of colors required to color the vertices of $G$ in such way that no path with an even number of vertices is comprised of two paths with the same color pattern. The construction presented above yields that the nonrepetitive chromatic number of every path with at least four vertices is three. The article answers in the positive the following question of Alon, Grytczuk, Hałuszczak and Riordan from 2002: Is the nonrepetitive chromatic number of planar graphs bounded? They show that the nonrepetitive chromatic number of every planar graph is at most 768 and provide generalizations to graphs embeddable to surfaces of higher genera and more generally to classes of graphs excluding a (topological) minor. Before their work, the best upper bound on the nonrepetitive chromatic number of planar graphs was logarithmic in their number of vertices, in addition to a universal upper bound quadratic in the maximum degree of a graph obtained using probabilistic method. The key ingredient for the argument presented in the article is the recent powerful result by Dujmović, Joret, Micek, Morin, Ueckerdt and Wood asserting that every planar graph is a subgraph of the strong product of a path and a graph of bounded tree-width (tree-shaped graph).

scholarly journals On $t$-Common List-Colorings

10.37236/6738 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Hojin Choi ◽  
Young Soo Kwon
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Planar Graphs ◽  
Four Color Theorem ◽  
List Colorings ◽  
List Chromatic Number ◽  
Positive Integers

In this paper, we introduce a new variation of list-colorings. For a graph $G$  and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) =  \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

scholarly journals A Note on Not-4-List Colorable Planar Graphs

10.37236/7320 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Margit Voigt ◽  
Arnfried Kemnitz
Keyword(s):  
Planar Graph ◽  
Bipartite Graph ◽  
Planar Graphs ◽  
The Other ◽  
Four Color Theorem ◽  
List Colorings ◽  
Vertex Set ◽  
Colorable Graph ◽  
Negative Strengthening

The Four Color Theorem states that every planar graph is properly 4-colorable. Moreover, it is well known that there are planar graphs that are non-$4$-list colorable. In this paper we investigate a problem combining proper colorings and list colorings. We ask whether the vertex set of every planar graph can be partitioned into two subsets where one subset induces a bipartite graph and the other subset induces a $2$-list colorable graph. We answer this question in the negative strengthening the result on non-$4$-list colorable planar graphs.

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 A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

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].

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.

An improved upper bound for the acyclic chromatic number of 1-planar graphs

2020 ◽  
Vol 283 ◽  
pp. 275-291
Author(s):  
Wanshun Yang ◽  
Weifan Wang ◽  
Yiqiao Wang
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs
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