scholarly journals Generalised Mycielski Graphs and the Borsuk–Ulam Theorem

10.37236/8462 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Tobias Müller ◽  
Matěj Stehlík
Keyword(s):  
Chromatic Number ◽  
Topological Method ◽  
Combinatorial Proof ◽  
Combinatorial Lemma ◽  
Ulam Theorem

Stiebitz determined the chromatic number of generalised Mycielski graphs using the topological method of Lovász, which invokes the Borsuk–Ulam theorem. Van Ngoc and Tuza used elementary combinatorial arguments to prove Stiebitz's theorem for 4-chromatic generalised Mycielski graphs, and asked if there is also an elementary combinatorial proof for higher chromatic number. We answer their question by showing that Stiebitz's theorem can be deduced from a version of Fan's combinatorial lemma. Our proof uses topological terminology, but is otherwise completely discrete and could be rewritten to avoid topology altogether. However, doing so would be somewhat artificial, because we also show that Stiebitz's theorem is equivalent to the Borsuk–Ulam theorem.

scholarly journals Local chromatic number and topology

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gábor Simonyi ◽  
Gábor Tardos
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Topological Method ◽  
Proper Coloring ◽  
And Topology ◽  
Minimum Number ◽  
International Audience ◽  
Closed Neighborhood

International audience The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.

scholarly journals On the Chromatic Number of Matching Kneser Graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 1-21
Author(s):  
Meysam Alishahi ◽  
Hajiabolhassan Hossein
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Large Family ◽  
Permutation Graphs ◽  
Kneser Graph ◽  
Ulam Theorem ◽  
Turán Number ◽  
Vertex Set ◽  
Kneser Graphs ◽  
Host Graph

AbstractIn an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.

scholarly journals Combinatorial Necklace Splitting

10.37236/168 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Recent Result ◽  
Combinatorial Proof ◽  
Ulam Theorem ◽  
Splitting Problem

We give a new, combinatorial proof for the necklace splitting problem for two thieves using only Tucker's lemma (a combinatorial version of the Borsuk-Ulam theorem). We show how this method can be applied to obtain a related recent result of Simonyi and even generalize it.

Packing Chromatic Number of Cycle Related Graphs

2014 ◽  
Vol 4 (1) ◽  
pp. 27
Author(s):  
Albert William ◽  
Roy Santiago ◽  
Indra Rajasingh
Keyword(s):  
Chromatic Number ◽  
Packing Chromatic Number

Packing coloring on subdivision-vertex and subdivision-edge join of cycle Cm with path Pn

2020 ◽  
pp. 627-634
Author(s):  
K. Rajalakshmi ◽  
M. Venkatachalam ◽  
M. Barani ◽  
D. Dafik
Keyword(s):  
Chromatic Number ◽  
Path Graph ◽  
Packing Chromatic Number ◽  
Cycle Graph

The packing chromatic number $\chi_\rho$ of a graph $G$ is the smallest integer $k$ for which there exists a mapping $\pi$ from $V(G)$ to $\{1,2,...,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. In this paper, the authors find the packing chromatic number of subdivision vertex join of cycle graph with path graph and subdivision edge join of cycle graph with path graph.

Adsorption of carbon monoxide on the iridium fcc (112) surface: Topological study

1989 ◽  
Vol 54 (3) ◽  
pp. 566-571 ◽  
Author(s):  
Jiří Pancíř ◽  
Ivana Haslingerová
Keyword(s):  
Carbon Monoxide ◽  
Topological Method ◽  
Dissociative Chemisorption ◽  
Semi Empirical

A semi-empirical topological method was applied to a study of an Ir(112) surface as well as to both a nondissociative and dissociative chemisorption on this surface. In all cases studied an attachment of carbon to the surface is energetically more favorable than an attachment of oxygen. The preferential capture of the CO molecule on atop sites is remarkable. The capture on n-fold hollow positions as well as the dissociative chemisorption of carbon monoxide on the Ir(112) surface are energetically prohibited.

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