8 research outputs found
Ramsey numbers of color critical graphs versus large generalized fans
Given two graphs and , the {Ramsey number} is the smallest
positive integer such that every 2-coloring of the edges of
contains either a red or a blue . Let be the
graph obtained from by adding a new vertex connecting
vertices of . Hook and Isaak (2011) defined the {\em star-critical
Ramsey number} as the smallest integer such that every
2-coloring of the edges of contains either a red or
a blue , where . For sufficiently large , Li and
Rousseau~(1996) proved that , Hao, Lin~(2018)
showed that ;
Li and Liu~(2016) proved that , and Li, Li,
and Wang~(2020) showed that . A graph
with is called edge-critical if contains an edge such
that . In this paper, we extend the above results by showing that
for an edge-critical graph with , when ,
and is sufficiently large, and
.Comment: 10 page
Ramsey Goodness and Beyond
In a seminal paper from 1983, Burr and Erdos started the systematic study of
Ramsey numbers of cliques vs. large sparse graphs, raising a number of
problems. In this paper we develop a new approach to such Ramsey problems using
a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type
stability, and other structural results. We give exact Ramsey numbers for
various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde
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 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
SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS
We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite
element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is
accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement