133,625 research outputs found
Classifying Families of Character Degree Graphs of Solvable Groups
We investigate prime character degree graphs of solvable groups. In
particular, we consider a family of graphs constructed by
adjoining edges between two complete graphs in a one-to-one fashion. In this
paper we determine completely which graphs occur as the prime
character degree graph of a solvable group.Comment: 7 pages, 5 figures, updated version is reorganize
What is Ramsey-equivalent to a clique?
A graph G is Ramsey for H if every two-colouring of the edges of G contains a
monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every
graph G is Ramsey for H if and only if it is Ramsey for H'. In this paper, we
study the problem of determining which graphs are Ramsey-equivalent to the
complete graph K_k. A famous theorem of Nesetril and Rodl implies that any
graph H which is Ramsey-equivalent to K_k must contain K_k. We prove that the
only connected graph which is Ramsey-equivalent to K_k is itself. This gives a
negative answer to the question of Szabo, Zumstein, and Zurcher on whether K_k
is Ramsey-equivalent to K_k.K_2, the graph on k+1 vertices consisting of K_k
with a pendent edge.
In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph
H if it is Ramsey for H but no proper subgraph of G is Ramsey for H. Let s(H)
be the smallest minimum degree over all Ramsey minimal graphs for H. The study
of s(H) was introduced by Burr, Erdos, and Lovasz, where they show that
s(K_k)=(k-1)^2. We prove that s(K_k.K_2)=k-1, and hence K_k and K_k.K_2 are not
Ramsey-equivalent.
We also address the question of which non-connected graphs are
Ramsey-equivalent to K_k. Let f(k,t) be the maximum f such that the graph
H=K_k+fK_t, consisting of K_k and f disjoint copies of K_t, is
Ramsey-equivalent to K_k. Szabo, Zumstein, and Zurcher gave a lower bound on
f(k,t). We prove an upper bound on f(k,t) which is roughly within a factor 2 of
the lower bound
On a theorem of K T Chen
Proving normal form of mappings of real line into itself by contracting mapping principl
Track Layouts of Graphs
A \emph{-track layout} of a graph consists of a (proper) vertex
-colouring of , a total order of each vertex colour class, and a
(non-proper) edge -colouring such that between each pair of colour classes
no two monochromatic edges cross. This structure has recently arisen in the
study of three-dimensional graph drawings. This paper presents the beginnings
of a theory of track layouts. First we determine the maximum number of edges in
a -track layout, and show how to colour the edges given fixed linear
orderings of the vertex colour classes. We then describe methods for the
manipulation of track layouts. For example, we show how to decrease the number
of edge colours in a track layout at the expense of increasing the number of
tracks, and vice versa. We then study the relationship between track layouts
and other models of graph layout, namely stack and queue layouts, and geometric
thickness. One of our principle results is that the queue-number and
track-number of a graph are tied, in the sense that one is bounded by a
function of the other. As corollaries we prove that acyclic chromatic number is
bounded by both queue-number and stack-number. Finally we consider track
layouts of planar graphs. While it is an open problem whether planar graphs
have bounded track-number, we prove bounds on the track-number of outerplanar
graphs, and give the best known lower bound on the track-number of planar
graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as
Technical Report TR-2003-07, School of Computer Science, Carleton University,
Ottawa, Canad
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