24 research outputs found
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
An updated survey on rainbow connections of graphs - a dynamic survey
The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow -connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study
Partitioning 2-edge-colored graphs by monochromatic paths and cycles
We present results on partitioning the vertices of -edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every -edge-colored graph
can be partitioned into at most monochromatic cycles, where
denotes the independence number of . Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from vertices, the vertex
set of any -edge-colored graph with minimum degree at least
(1+\eps){3|V(G)|\over 4} can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that does not contain a fixed bipartite graph , we show that
in every -edge-coloring of , vertices can be covered by two
vertex disjoint paths of different colors, where is a constant depending
only on . In particular, we prove that , which is best possible