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Remarks on 15-vertex (3,3)-ramsey graphs not containing K₅

100%
Urbański S.
Discussiones Mathematicae Graph Theory
|
1996
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tom 16
|
nr 2
173-179
EN
The paper gives an account of previous and recent attempts to determine the order of a smallest graph not containing K₅ and such that every 2-coloring of its edges results in a monochromatic triangle. A new 14-vertex K₄-free graph with the same Ramsey property in the vertex coloring case is found. This yields a new construction of one of the only two known 15-vertex (3,3)-Ramsey graphs not containing K₅.
2
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A ramsey-type theorem for multiple disjoint copies of induced subgraphs

100%
Nakamigawa T.
Discussiones Mathematicae Graph Theory
|
2014
|
tom 34
|
nr 2
249-261
EN
Let k and ℓ be positive integers with ℓ ≤ k − 2. It is proved that there exists a positive integer c depending on k and ℓ such that every graph of order (2k−1−ℓ/k)n+c contains n vertex disjoint induced subgraphs, where these subgraphs are isomorphic to each other and they are isomorphic to one of four graphs: (1) a clique of order k, (2) an independent set of order k, (3) the join of a clique of order ℓ and an independent set of order k − ℓ, or (4) the union of an independent set of order ℓ and a clique of order k − ℓ.
3
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Pancyclicity when each Cycle Must Pass Exactly k Hamilton Cycle Chords

100%
Affif Chaouche F., Rutherford C. G., Whitty R. W.
Discussiones Mathematicae Graph Theory
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2015
|
tom 35
|
nr 3
533-539
EN
It is known that Θ(log n) chords must be added to an n-cycle to produce a pancyclic graph; for vertex pancyclicity, where every vertex belongs to a cycle of every length, Θ(n) chords are required. A possibly ‘intermediate’ variation is the following: given k, 1 ≤ k ≤ n, how many chords must be added to ensure that there exist cycles of every possible length each of which passes exactly k chords? For fixed k, we establish a lower bound of ∩(n1/k) on the growth rate.
4
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Vertex-disjoint copies of K¯₄

100%
Kawarabayashi K.-i.
Discussiones Mathematicae Graph Theory
|
2004
|
tom 24
|
nr 2
249-262
EN
Let G be a graph of order n. Let K¯ₗ be the graph obtained from Kₗ by removing one edge. In this paper, we propose the following conjecture: Let G be a graph of order n ≥ lk with δ(G) ≥ (n-k+1)(l-3)/(l-2)+k-1. Then G has k vertex-disjoint K¯ₗ. This conjecture is motivated by Hajnal and Szemerédi's [6] famous theorem. In this paper, we verify this conjecture for l=4.
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