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On invariants related to edge numbers of Kl+1-free graphs
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0002-6188-0596
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we discuss a generalisation of Turán's theorem, where we do not only consider an upper bound on the clique number but also on the independence number, α(G). This generalisation can be expressed in terms of the non-negativity of the graph invariant

iₗ(G) = 2e(G) - ⌈(7\ell-2)/2⌉ n(G) + l ⌊(5\ell + 1)/2⌋ α(G),

for all Kₗ₊₁-free graphs G, where e(G) and n(G) denote the number or edges and the number of vertices of the graph G, respectively. We show several strong properties that must be satisfied by a minimal Kₗ₊₁-free graph such that iₗ(G) < 0. Moreover, for some special cases for l we show that indeed iₗ(G) ≥ 0 and classify the graphs for which we have equality.

National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-174784OAI: oai:DiVA.org:su-174784DiVA, id: diva2:1359784
Available from: 2019-10-10 Created: 2019-10-10 Last updated: 2019-10-25Bibliographically approved
In thesis
1. On linear graph invariants related to Ramsey and edge numbers: or how I learned to stop worrying and love the alien invasion
Open this publication in new window or tab >>On linear graph invariants related to Ramsey and edge numbers: or how I learned to stop worrying and love the alien invasion
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study the Ramsey numbers, R(l,k), the edge numbers, e(l,k;n) and graphs that are related to these. The edge number e(l,k;n) may be defined as the least natural number m for which all graphs on n vertices and less than m edges either contains a complete subgraph of size l or an independent set of size k. The Ramsey number R(l,k) may then be defined as the least natural number n for which e(l,k;n) = ∞ .

In Paper I, IV and V we study strict lower bounds for e(l,k;n). In Paper I we do this in the case where l = 3 by, in particular, showing e(G) ≥ (1/3)(17n(G) - 35α(G) - N(C4;G)) for all triangle-free graphs G, where N(C4;G) denotes the number of cycles of length 4 in G. In Paper IV we describe a general method for generating similar inequalities to the one above but for graphs that may contain triangles, but no complete subgraphs of size 4. We then show a selection of the inequalities we get from the computerised generation. In Paper V we study the inequality 

e(G) ≥ (1/2)(ceil((7l - 2)/2)n(G) - l floor((5l + 1)/2)α(G))

for l ≥ 2, and examine the properties of graphs G without cliques of size l+1 such that G is minimal with respect to the above inequality not holding, and show for small l that no such graphs G can exist.

In Paper II we study constructions of graphs G such that e(G) - e(3,k;n) is small when n ≤ 3.5(k-1). We employ a description of some of these graphs in terms of 'patterns' and a recursive procedure to construct them from the patterns. We also present the result of computer calculations where we actually have performed such constructions of Ramsey graphs and compare these lists to previously computed lists of Ramsey graphs.

In Paper III we develop a method for computing, recursively, upper bounds for Ramsey numbers R(l,k). In particular the method uses bounds for the edge numbers e(l,k;n). In Paper III we have implemented this method as a computer program which we have used to improve several of the best known upper bounds for small Ramsey numbers R(l,k).

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2019. p. 47
Keywords
Ramsey number, edge number, minimal Ramsey graph, independence number, clique number, Turán's theorem, crochet pattern, H13-pattern, linear graph invariant, triangle-free graph
National Category
Mathematics Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-174786 (URN)978-91-7797-905-0 (ISBN)978-91-7797-906-7 (ISBN)
Public defence
2019-12-16, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Accepted. Paper 4: Manuscript. Paper 5: Manuscript.

Available from: 2019-11-21 Created: 2019-10-10 Last updated: 2019-11-19Bibliographically approved

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