1,256 research outputs found
THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References
and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a
Some exact values on Ramsey numbers related to fans
For two given graphs and , the Ramsey number is the smallest
integer such that any red-blue edge-coloring of the complete graph
contains a red or a blue . When , we simply write . For an
positive integer , let be a star with vertices, be a
fan with vertices consisting of triangles sharing one common vertex,
and be a graph with vertices obtained from the disjoint union of
triangles. In 1975, Burr, Erd\H{o}s and Spencer \cite{B} proved that
for . However, determining the exact value of
is notoriously difficult. So far, only has been proved. Notice
that both and contain triangles and for
all . Chen, Yu and Zhao (2021) speculated that for sufficiently large. In this paper, we first prove that
for , where if is
odd and if is even. Applying the exact values of
, we will confirm for by showing that
.Comment: 10 pages, 3 figure
The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths
The Ramsey number for a pair of graphs and is defined as the smallest integer such that, for any graph on vertices, either contains or contains as a subgraph, where denotes the complement of . We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers and for some integers , , where is a linear forest of order with at least one edge
Thermal atmospheric models
The static thermal atmosphere is described and its predictions are compared to observations both to test the validity of the classic assumptions and to distinguish and describe those spectral features with diagnostic value
Small Ramsey Numbers
We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values
Cycle-complete Ramsey numbers
The Ramsey number is the smallest natural number such
that every red/blue edge-colouring of a clique of order contains a red
cycle of length or a blue clique of order . In 1978, Erd\H{o}s,
Faudree, Rousseau and Schelp conjectured that for provided .
We prove that, for some absolute constant , we have provided . Up to the
value of this is tight since we also show that, for any
and , we have
for all .
This proves the conjecture of Erd\H{o}s, Faudree, Rousseau and Schelp for
large , a stronger form of the conjecture due to Nikiforov, and answers
(up to multiplicative constants) two further questions of Erd\H{o}s, Faudree,
Rousseau and Schelp.Comment: 19 page
Cycle-complete ramsey numbers
The Ramsey number r(Cℓ, Kn) is the smallest natural number N such that every red/blue edge-colouring of a clique of order N contains a red cycle of length ℓ or a blue clique of order n. In 1978, Erdos, Faudree, Rousseau and Schelp conjectured that r(Cℓ, Kn) = (ℓ − 1)(n − 1) + 1 for ℓ ≥ n ≥ 3 provided (ℓ, n) 6= (3, 3). We prove that, for some absolute constant C ≥ 1, we have r(Cℓ, Kn) = (ℓ − 1)(n − 1) + 1 provided ℓ ≥ C logloglognn. Up to the value of C this is tight since we also show that, for any ε > 0 and n > n0(ε), we have r(Cℓ, Kn) ≫ (ℓ − 1)(n − 1) + 1 for all 3 ≤ ℓ ≤ (1 − ε)logloglognn. This proves the conjecture of Erdos, Faudree, Rousseau and Schelp for large ℓ, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erdos, Faudree, Rousseau and Schelp
Ramsey goodness of books revisited
The Ramsey number is the minimum such that every graph on
vertices contains as a subgraph or its complement contains as a
subgraph. For integers , the -book is the graph
on vertices consisting of a copy of , called the spine, as well as
additional vertices each adjacent to every vertex of the spine and
non-adjacent to each other. A connected graph on vertices is called
-good if . Nikiforov and Rousseau proved that if
is sufficiently large in terms of and , then is -good.
Their proof uses Szemer\'edi's regularity lemma and gives a tower-type bound on
. We give a short new proof that avoids using the regularity method and
shows that every with is -good.
Using Szemer\'edi's regularity lemma, Nikiforov and Rousseau also proved much
more general goodness-type results, proving a tight bound on for
several families of sparse graphs and as long as for a small constant . Using our techniques, we prove a new
result of this type, showing that when and
is a complete -partite graph whose first parts have constant size
and whose last part has size , for some small constant .
Again, our proof does not use the regularity method, and thus yields
double-exponential bounds on .Comment: 21 page
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