30 research outputs found
On the minimum degree of minimal Ramsey graphs for multiple colours
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every
r-colouring of the edges of G contains a monochromatic copy of H. The graph G
is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph
of G possesses this property. Let s_r(H) denote the smallest minimum degree of
G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter
s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed
that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of
s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) =
r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in
both r and k, and we determine s_r(K_3) up to a factor of log r
On the minimum degree of minimal Ramsey graphs
We investigate the minimization problem of the minimum degree of minimal Ramsey graphs, initiated by Burr, ErdĹs, and LovĂĄsz. We determine the corresponding graph parameter for numerous bipartite graphs, including bi-regular bipartite graphs and forests. We also make initial progress for graphs of larger chromatic number. Numerous interesting problems remain open
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
Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available