graph minor
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2020 ◽  
Vol 30 (8) ◽  
pp. 952-975
Author(s):  
Anton Freund

AbstractHarvey Friedman’s gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: It arises via iterated applications of a uniform Kruskal theorem.


2017 ◽  
Vol 16 (5) ◽  
Author(s):  
Arman Zaribafiyan ◽  
Dominic J. J. Marchand ◽  
Seyed Saeed Changiz Rezaei

10.37236/5715 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
David R. Wood

This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.


2015 ◽  
Vol 24 (13) ◽  
pp. 1541004
Author(s):  
Vassily Olegovich Manturov

In the present paper, we proceed with the study of framed 4-graph minor theory initiated in [V. O. Manturov, Framed 4-valent graph minor theory I: Intoduction planarity criterion, arxiv: 1402.1564v1 [Math.Co]] and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in [Formula: see text].


2014 ◽  
Vol 60 (9) ◽  
pp. 5375-5386 ◽  
Author(s):  
Xunrui Yin ◽  
Yan Wang ◽  
Zongpeng Li ◽  
Xin Wang ◽  
Xiangyang Xue
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2014 ◽  
Vol 23 (07) ◽  
pp. 1460002
Author(s):  
Vassily Olegovich Manturov

This paper is the first one in the sequence of papers about a simple class of framed 4-graphs; the goal of this paper is to collect some well-known results on planarity and to reformulate them in the language of minors. The goal of the whole sequence is to prove analogs of the Robertson–Seymour–Thomas theorems for framed 4-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed 4-graphs are easier to consider than general graphs; on the other hand, framed 4-graphs are closely related to many problems in graph theory.


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