Richard H. Schelp

All Ramsey numbers for cycles in graphs

In the past, Ramsey numbers were known for pairs of cycles of lengths r and s when one of the following occurred: (1) r and s are small, (2) one of r or s is small relative to the other, or (3) r is odd and r = s. In this paper we complete the Ramsey number problem for cycles by verifying their previously conjectured values.

Vertex-distinguishing proper edge-colorings

An edge-coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge-coloring of a simple graph G is denoted by . A simple count shows that where ni denotes the number of vertices of degree i in G. We prove that where C is a constant depending only on Δ. Some results for special classes of graphs, notably trees, are also presented.

THE SIZE RAMSEY NUMBER

Let denote the class of all graphsG which satisfyG→(G1,G2). As a way of measuring minimality for members of, we define thesize Ramsey number ř(G1,G2) by.We then investigate various questions concerned with the asymptotic behaviour ofř.

Adjacent Vertex Distinguishing Edge-Colorings

An adjacent vertex distinguishing edge-coloring of a simple graph

Graphs with linearly bounded Ramsey numbers

G

Neighborhood unions and hamiltonian properties in graphs

is a proper edge-coloring of

Vertex distinguishing colorings of graphs with Δ(G)=2

G

On cycle—Complete graph ramsey numbers

such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors

Path Ramsey numbers in multicolorings

\chi^\prime_a(G)

Ramsey numbers for local colorings

required to give