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On Ramsey Minimal Graphs
1994
Electronic Journal of Combinatorics
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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. ...
doi:10.37236/1184
fatcat:id3vugm6g5hjvlnstdq3yrnhze
On Ramsey minimal graphs
2004
Discrete Mathematics
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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. ...
doi:10.1016/j.disc.2003.11.043
fatcat:alk4j5cuc5c5bnwn7apa6o4cuy
On Ramsey-minimal infinite graphs
[article]
2021
arXiv
pre-print
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2020 pre-print arXiv:2011.14074v1 fatcat:afyy5uu3rbbw5lsiymeul2waciOther Versions
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] . ...
arXiv:2011.14074v2
fatcat:fmrykone2jhlrkpnusf3oigbve
On Ramsey-Minimal Infinite Graphs
2021
Electronic Journal of Combinatorics
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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. ...
doi:10.37236/10046
fatcat:ykzprww74jahvlfmqbvgq645ae
On Ramsey (K1,2,C4)-minimal graphs
2010
Discussiones Mathematicae Graph Theory
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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. ...
doi:10.7151/dmgt.1519
fatcat:osjhiuw46faetpirtoqpkjhce4
On minimal Ramsey graphs and Ramsey equivalence in multiple colours
[article]
2018
arXiv
pre-print
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(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. ...
arXiv:1809.09232v1
fatcat:aqcpcxcsgra45bxelyfzcgj6ya
On Unicyclic Ramsey (mK2, P3)−Minimal Graphs
2015
Procedia Computer Science
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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. ...
doi:10.1016/j.procs.2015.12.067
fatcat:4zhgiwgxpbfjteohqfh7ft6qja
On the minimum degree of minimal Ramsey graphs
2009
Journal of Graph Theory
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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. ...
doi:10.1002/jgt.20445
fatcat:rtybkrkswnbyxopfuofz4mhr7u
On Ramsey (K_{1,2},K_{n})-minimal graphs
2012
Discussiones Mathematicae Graph Theory
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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). ...
doi:10.7151/dmgt.1604
fatcat:ycysz55v7feita6en5w4sr7eea
On Ramsey Minimal Graphs for the Pair Paths
2015
Procedia Computer Science
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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. ...
doi:10.1016/j.procs.2015.12.068
fatcat:jjz6wnr32zg4jkhsh7dq3zbyee
On the minimum degree of minimal Ramsey graphs for multiple colours
[article]
2015
arXiv
pre-print
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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 . ...
arXiv:1502.02881v1
fatcat:djpcaevlunafhabm47tfunkkvu
On the minimum degree of minimal Ramsey graphs for multiple colours
2016
Journal of combinatorial theory. Series B (Print)
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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. ...
doi:10.1016/j.jctb.2016.03.006
fatcat:jjybf7twvvfknixguq6pweh6ni
Vertex Folkman Numbers and the Minimum Degree of Minimal Ramsey Graphs
2018
SIAM Journal on Discrete Mathematics
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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 . ...
doi:10.1137/17m1116696
fatcat:e62ftv7r7ndu5otoi3trx3ki4q
Ramsey minimal graphs for a pair of a cycle on four vertices and an arbitrary star
2022
Electronic Journal of Graph Theory and Applications
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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] . ...
doi:10.5614/ejgta.2022.10.1.20
fatcat:vmm7ph4pwba7ngdiprv5c7z7ai
On Ramsey (mK2, P4)-Minimal Graphs
2022
Advances in Computer Science Research
unpublished
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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 , ...
doi:10.2991/acsr.k.220202.003
fatcat:f6riedyswvhw7pwg5ahmpyw6de
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