Ramsey numbers of color critical graphs versus large generalized fans

Given two graphs $G$ and $H$, the {Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that every 2-coloring of the edges of $K_{N}$ contains either a red $G$ or a blue $H$. Let $K_{N-1}\sqcup K_{1,k}$ be the graph obtained from $K_{N-1}$ by adding a new vertex $v$ connecting $k$ vertices of $K_{N-1}$. Hook and Isaak (2011) defined the {\em star-critical Ramsey number} $r_{*}(G,H)$ as the smallest integer $k$ such that every 2-coloring of the edges of $K_{N-1}\sqcup K_{1,k}$ contains either a red $G$ or a blue $H$, where $N=R(G, H)$. For sufficiently large $n$, Li and Rousseau~(1996) proved that $R(K_{k+1},K_{1}+nK_{t})=knt +1$, Hao, Lin~(2018) showed that $r_{*}(K_{k+1},K_{1}+nK_{t})=(k-1)tn+t$; Li and Liu~(2016) proved that $R(C_{2k+1}, K_{1}+nK_{t})=2nt+1$, and Li, Li, and Wang~(2020) showed that $r_{*}(C_{2m+1},K_{1}+nK_{t})=nt+t$. A graph $G$ with $\chi(G)=k+1$ is called edge-critical if $G$ contains an edge $e$ such that $\chi(G-e)=k$. In this paper, we extend the above results by showing that for an edge-critical graph $G$ with $\chi(G)=k+1$, when $k\geq 2$, $t\geq 2$ and $n$ is sufficiently large, $R(G, K_{1}+nK_{t})=knt+1$ and $r_{*}(G,K_{1}+nK_{t})=(k-1)nt+t$.Comment: 10 page

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

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

Bowdoin Orient v.133, no.1-24 (2003-2004)

https://digitalcommons.bowdoin.edu/bowdoinorient-2000s/1004/thumbnail.jp

A Holmes and Doyle Bibliography, Volume 6: Periodical Articles, Subject Listing, By De Waal Category

This bibliography is a work in progress. It attempts to update Ronald B. De Waal’s comprehensive bibliography, The Universal Sherlock Holmes, but does not claim to be exhaustive in content. New works are continually discovered and added to this bibliography. Readers and researchers are invited to suggest additional content. Volume 6 presents the periodical literature arranged by subject categories (as originally devised for the De Waal bibliography and slightly modified here)