scholarly journals Random Planar Maps and Graphs with Minimum Degree Two and Three

10.37236/7640 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Marc Noy ◽  
Lander Ramos
Keyword(s):  
Planar Graph ◽  
Random Graphs ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Asymptotic Estimates ◽  
The Core ◽  
New Information ◽  
Planar Maps ◽  
Explicit Constant ◽  
Precise Asymptotic

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order $c \log(n)$ for an explicit constant $c$. These results provide new information on the structure of random planar 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 Induced 2-Degenerate Subgraphs of Triangle-Free Planar Graphs

10.37236/7311 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Tom Kelly
Keyword(s):  
Planar Graph ◽  
Lower Bounds ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Euler’S Formula

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$.  We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph.  We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $\alpha_2(G)$.  We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $\alpha_2(G) \geq \frac{6n - m - 1}{5}$.  By Euler's Formula, this implies $\alpha_2(G) \geq \frac{4}{5}n$.  We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most three, then $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.

The structure of plane graphs with independent crossings and its applications to coloring problems

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Xin Zhang ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Plane Graph ◽  
Plane Graphs ◽  
Total Chromatic Number ◽  
Linear Arboricity

AbstractIf a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.

scholarly journals New Upper Bounds for the Heights of Some Light Subgraphs in 1-Planar Graphs with High Minimum Degree

2011 ◽  
Vol Vol. 13 no. 3 (Combinatorics) ◽  
Author(s):  
Xin Zhang ◽  
Jian-Liang Wu ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Upper Bounds ◽  
The Family ◽  
International Audience

Combinatorics International audience A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph of minimum degree 6 contains a copy of 4-cycle with all vertices of degree at most 19. In addition, we also show that the complete graph K 4 is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.

scholarly journals Presuppositions, implicatures, and contextual equivalence

2021 ◽  
Author(s):  
Paul Marty ◽  
Jacopo Romoli
Keyword(s):  
Linguistic Theory ◽  
Empirical Adequacy ◽  
The Novel ◽  
Systematic Comparison ◽  
The Core ◽  
New Information ◽  
Maximize Presupposition ◽  
Important Challenge ◽  
Contextual Equivalence

AbstractMaximize Presupposition! (MP), as originally proposed in Heim (Semantik: Ein internationales Handbuch der zeitgenössischen Forschung, pp. 487–535, 1991) and developed in subsequent works, offers an account of the otherwise mysterious unassertability of a variety of sentences. At the core of MP is the idea that speakers are urged to use a sentence ψ over a sentence ϕ if ψ contributes the same new information as ϕ, yet carries a stronger presupposition. While MP has been refined in many ways throughout the years, most (if not all) of its formulations have retained this characterisation of the MP-competition. Recently, however, the empirical adequacy of this characterisation has been questioned in light of certain newly discovered cases that are infelicitous, despite meeting MP-competition conditions. This has led some researchers to broaden the scope of MP, extending it to competition between sentences which are not contextually equivalent (Spector and Sudo in Linguistics and Philosophy 40(5):473–517, 2017) and whose presuppositions are not satisfied in the context (Anvari in Proceedings of Semantics and Linguistic Theory 28, pp. 711–726, 2018; Manuscript, IJN-ENS, 2019). In this paper, we present a body of evidence showing that these formulations of MP are sometimes too liberal, sometimes too restrictive: they overgenerate infelicity for a variety of felicitous cases while leaving the infelicity of minimally different cases unaccounted for. We propose an alternative, implicature-based approach stemming from Magri (PhD dissertation, MIT, 2009), Meyer (PhD dissertation, MIT, 2013), and Marty (PhD dissertation, MIT, 2017), which reintroduces contextual equivalence and presupposition satisfaction in some form through the notion of relevance. This approach is shown to account for the classical and most of the novel cases. Yet some of the latter remain problematic for this approach as well. We end the paper with a systematic comparison of the different approaches to MP and MP-like phenomena, covering both the classical and the novel cases. All in all, the issue of how to properly restrict the competition for MP-like phenomena remains an important challenge for all accounts in the literature.

Lexical Arguments and Information Types in Malay Oral Narratives

2015 ◽  
Vol 7 (1) ◽  
pp. 80-97
Author(s):  
Siaw-Fong Chung
Keyword(s):  
Information Flow ◽  
The Other ◽  
Discourse Structure ◽  
Oral Narratives ◽  
Clear Cut ◽  
The Core ◽  
New Information ◽  
Intonation Units ◽  
Locative Expressions ◽  
Transitive Verbs

The analysis in this paper was based on five Malay narratives of the “frog story”. In these narratives, the types of lexical arguments and their relations with information flow and topic continuity were analyzed. It was found that most narrators used one lexical argument in telling the frog story (e.g., sarang itu jatuh “the nest fell”). About 60% of the verbs in the narratives contained one lexical argument only. Some transitive verbs that usually require the presence of both lexical arguments were used with one lexical argument only when produced in speech (e.g., dia mencari ø di merata tempat “he searched (for) ø everywhere”). Objects were sometimes omitted, as their meanings could be predicted from previous context. Despite the omission of objects, transitive constructions still prevailed in the stories. The most frequently occurring lexical arguments were objects (O) (37%), followed by intransitive subjects (S) (29%) and transitive subjects (A) (27%). In addition, our results showed that new information in Malay was usually allocated to the core argument of the object and to locative expressions, indicating that most of the new information appeared at the end of a clause. On the other hand, topic continuity was held between the subjects in two continuous intonation units. This clear-cut division of discourse functions in the heads and tails of constructions was consistently found in the five pieces of narration. This observation not only showed how ideas could be continued in Malay oral narratives, but also contributes to the study of discourse structure in Malay.

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 Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

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