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### On Ramsey Minimal Graphs

Tomasz Łuczak
1994 Electronic Journal of Combinatorics
An elementary probabilistic argument is presented which shows that for every forest $F$ other than a matching, and every graph $G$ containing a cycle, there exists an infinite number of graphs $J$ such  ...  that $J\to (F,G)$ but if we delete from $J$ any edge $e$ the graph $J-e$ obtained in this way does not have this property.  ...  Let F be any forest on m vertices, other than a matching, and let G be a graph containing at least one cycle.  ...

### On Ramsey minimal graphs

Mieczysław Borowiecki, Mariusz Hałuszczak, Elżbieta Sidorowicz
2004 Discrete Mathematics
The graph G is (F; H )-minimal (Ramsey-minimal) if G → (F; H ) but G 9 (F; H ) for any proper subgraph G ⊆ G. The class of all (F; H )-minimal graphs will be denoted by R(F; H ).  ...  To prove the ÿrst one we will use the well-known result which describes the graphs with a 1-factor.  ...

### On Ramsey-minimal infinite graphs [article]

Jordan Mitchell Barrett, Valentino Vito
2021 arXiv   pre-print
We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair (G,H) to have a Ramsey-minimal graph.  ...  We use these to prove, for example, that if G=S_∞ is an infinite star and H=nK_2, n ≥ 1 is a matching, then the pair (S_∞,nK_2) admits no Ramsey-minimal graphs.  ...  For an example of previous work on Ramsey-minimal finite graphs involving matchings, see Burr et al. [4] .  ...

### On Ramsey-Minimal Infinite Graphs

Jordan Barrett, Valentino Vito
2021 Electronic Journal of Combinatorics
We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph.  ...  We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.  ...  Also, we would like to thank the anonymous referees for their valuable feedback and comments on this paper.  ...

### On Ramsey (K1,2,C4)-minimal graphs

Edy Tri Baskoro, Tomáš Vetrik, Lyra Yulianti
2010 Discussiones Mathematicae Graph Theory
is a Ramsey (K 1,2 , C 4 )-minimal graph.  ...  If F is Ramsey (G(U 0 ) 0 , H)-minimal, we write F ∈ R(G(U 0 ), H). Par- ticularly, for U 0 = ∅, F is a Ramsey (G, H)-minimal graph.  ...

### On minimal Ramsey graphs and Ramsey equivalence in multiple colours [article]

Dennis Clemens, Anita Liebenau, Damian Reding
2018 arXiv   pre-print
(H) denotes the set of all graphs that are q-Ramsey-minimal for H.  ...  For such H, the following are some consequences. (1) For 2< r< q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H. (2) For  ...  Clearly, every graph G that is a q-minimal graph for some graph H is r-Ramsey for H, for all 2 ď r ď q, and thus contains an r-minimal graph as an induced subgraph.  ...

### On Unicyclic Ramsey (mK2, P3)−Minimal Graphs

Kristiana Wijaya, Edy Tri Baskoro, Hilda Assiyatun, Djoko Suprijanto
2015 Procedia Computer Science
Ramsey minimal graph is one of growing topics in Ramsey theory. The search of Ramsey minimal graphs for a combination of graphs G and H is an interesting and difficult problem.  ...  In this paper, we study the properties of Ramsey minimal graphs for G = mK 2 and H = P n . In particular, we determine all unicyclic Ramsey minimal graphs for this combination.  ...  A graph F is said to be a Ramsey (G, H)−minimal if F → (G, H) but for each e ∈ E(F), (F − e) (G, H). Denote by R(G, H) the set of all Ramsey (G, H)−minimal graphs.  ...

### On the minimum degree of minimal Ramsey graphs

Tibor Szabó, Philipp Zumstein, Stefanie Zürcher
2009 Journal of Graph Theory
We investigate the minimization problem of the minimum degree of minimal Ramsey graphs, initiated by Burr, Erdős, and Lovász.  ...  We determine the corresponding graph parameter for numerous bipartite graphs, including bi-regular bipartite graphs and forests. We also make initial progress for graphs of larger chromatic number.  ...  One of them simplified the proofs of Theorems 1.7 and 3.1 considerably, also correcting an error in the latter.  ...

### On Ramsey (K_{1,2},K_{n})-minimal graphs

Mariusz Hałuszczak
2012 Discussiones Mathematicae Graph Theory
Let F be a graph and let G, H denote nonempty families of graphs.  ...  These graphs together with our previous results allow us to construct infinitely many (K 1,2 , K n )minimal graphs, i.e. graphs that belong to the Ramsey set ℜ(K 1,2 , K n ) for every n ≥ 3. Proof.  ...  The Ramsey set ℜ(G, H) is defined to be the set of all (G, H)-minimal graphs (up to isomorphism). For the simplicity of the notation, instead of ℜ({G}, {H}) we write ℜ(G, H).  ...

### On Ramsey Minimal Graphs for the Pair Paths

2015 Procedia Computer Science
Introduction We call a graph F as a Ramsey (G,H)-minimal graph if F →(G, H) but F * (G, H) for every F * ⊂ F.  ...  Finding all the Ramsey (G, H)-minimal graphs for particular G and H is a very interesting but difficult problem, even though for small graphs G and H.  ...

### On the minimum degree of minimal Ramsey graphs for multiple colours [article]

Jacob Fox, Andrey Grinshpun, Anita Liebenau, Yury Person, Tibor Szabo
2015 arXiv   pre-print
The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property.  ...  Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H.  ...  G on at most n vertices that are r-Ramsey-minimal for the clique K k .  ...

### On the minimum degree of minimal Ramsey graphs for multiple colours

Jacob Fox, Andrey Grinshpun, Anita Liebenau, Yury Person, Tibor Szabó
2016 Journal of combinatorial theory. Series B (Print)
A graph G is r-Ramsey for a graph H, denoted by G → (H) r , if every r-colouring of the edges of G contains a monochromatic copy of H.  ...  The graph G is called r-Ramseyminimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property.  ...  A graph G is r-Ramsey-minimal for H (or r-minimal for H) if G → (H) r , but G (H) r for any proper subgraph G G.  ...

### Vertex Folkman Numbers and the Minimum Degree of Minimal Ramsey Graphs

Hiệp Hàn, Vojtěch Rödl, Tibor Szabó
2018 SIAM Journal on Discrete Mathematics
We investigate the smallest possible minimum degree of r-color minimal Ramsey-graphs for the k-clique.  ...  As a side product our result also yields an improved upper bound on the vertex Folkman number F (r, k, k + 1) of the k-clique.  ...  A graph G is r-Ramsey-minimal for H (or r-minimal for H) if G → (H) r , but none of the proper subgraphs G G satisfies G → (H) r .  ...

### Ramsey minimal graphs for a pair of a cycle on four vertices and an arbitrary star

Maya Nabila, Hilda Assiyatun, Edy Tri Baskoro
2022 Electronic Journal of Graph Theory and Applications
In this paper, we give some finite and infinite classes of Ramsey (C 4 , K 1,n )-minimal graphs for any n ≥ 3.  ...  A graph F is called a Ramsey (G, H)-minimal graph if it satisfies two conditions: (i) F → (G, H) and (ii) F − e (G, H) for any edge e of F .  ...  The study of Ramsey minimal graphs was initiated by Burr et al. [3] .  ...

### On Ramsey (mK2, P4)-Minimal Graphs

Asep Iqbal Taufik, Denny Riama Silaban, Kristiana Wijaya
2022 Advances in Computer Science Research   unpublished
Let 𝑚𝐾 2 be matching with m edges and 𝑃 𝑛 be a path on n vertices. In this paper, we construct all disconnected Ramsey minimal graphs, and found some new connected graphs in ℛ(3𝐾 2 , 𝑃 4 ).  ...  Graph 𝐹 is a Ramsey (𝐺, 𝐻)-minimal if 𝐹 → (𝐺, 𝐻) but for each 𝑒 ∈ 𝐸(𝐹), (𝐹 − 𝑒) ↛ (𝐺, 𝐻). The set ℛ(𝐺, 𝐻) consists of all Ramsey (𝐺, 𝐻)-minimal graphs.  ...  [8] constructed a family of Ramsey (𝑚𝐾 2 , 𝑃 4 ) minimal graphs from Ramsey ((𝑚 − 1)𝐾 2 , 𝑃 4 ) minimal graph by doing 4 times subdivision on any edge belongs to a cycle in a Ramsey (𝑚𝐾 2 ,  ...
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