Table Of ContentConditions on Ramsey non-equivalence
Maria Axenovich, Jonathan Rollin, and Torsten Ueckerdt
5
1 Department of Mathematics, Karlsruhe Institute of Technology
0
2
March 25, 2015
r
a
M
Abstract
4
2 GivenagraphH,agraphGiscalledaRamsey graph of H ifthereisamonochro-
maticcopyofH ineverycoloringoftheedgesofGwithtwocolors. TwographsG,H
] are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al.
O
[J. Combin. Theory Ser. B 109 (2014), 120–133] asked whether there are two non-
C
isomorphicconnectedgraphsthatareRamseyequivalent. Theyprovedthatacliqueis
. notRamseyequivalenttoanyotherconnectedgraph. ResultsofNeˇsetˇriletal. showed
h
t thatanytwographswithdifferentcliquenumber[Combinatorica 1(2)(1981),199–202]
a or different odd girth [Comment. Math. Univ. Carolin. 20(3) (1979), 565–582] are
m
notRamseyequivalent. Thesearetheonlystructuralgraphparametersweknowthat
[ “distinguish” two graphs in the above sense. This paper provides further supportive
evidence for a negative answer to the question of Fox et al. by claiming that for wide
2
classes of graphs, chromatic number is a distinguishing parameter. In addition, it is
v
shownherethatallstarsandpathsandallconnectedgraphsonatmost5verticesare
0
2 notRamseyequivalenttoanyotherconnectedgraph. Moreovertwoconnectedgraphs
3 are not Ramsey equivalent if they belong to a special class of trees or to classes of
6 graphs with clique-reduction properties.
0
Keywords: Ramsey, Ramsey equivalence, chromatic number, Ramsey classes
.
1
0
5 1 Introduction
1
:
v Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic
i
X copy of H in every coloring ofthe edges of G with two colors. If G is a Ramsey graphof H,
we write G H and say that G arrows H. We denote by (H) the set of all graphs that
r → R
a arrow H, and call it the Ramsey class of H. So the Ramsey number R(H) is the smallest
integer n such that K (H), where K denotes the complete graph on n vertices. Two
n n
∈R
graphs G, H are called Ramsey equivalent if they have the same Ramsey class. We write
R R
G H if G is Ramsey equivalent to H, and write G H otherwise. The study of Ramsey
∼ (cid:54)∼
classeswasinitiatedbythefundamentalworkofNeˇsetˇrilandR¨odl[22]andBurr,Erd˝os,and
Lov´asz [8]. However, the notion of Ramsey equivalence of graphs was raised only recently
by Szab´o et al. [28]. It was shown in [28], that there are two non-isomorphic graphs that
R
are Ramsey equivalent, for example G H, where G=K and H is a vertex disjoint union
t
∼
of K and K for t 4. In this example H is a disconnected graph. Fox et al. formulated
t 2
≥
a question for connected graphs:
1
Question 1 ([11]). Are there two non-isomorphic connected graphs G and H with G R H?
∼
R
Note that G H if and only if there exists a graph Γ such that Γ H and Γ G or
(cid:54)∼ → (cid:54)→
Γ G and Γ H. In this case we call Γ a graph, distinguishing G and H. So in order
→ (cid:54)→
R
to prove that G H, it is sufficient to explicitly construct a distinguishing graph. Another
(cid:54)∼
R
approach is to identify a graph parameter s, such that s(G) = s(H) implies that G H.
(cid:54) (cid:54)∼
In this case, we say that s is a Ramsey distinguishing parameter. The only structural graph
parameters that we know to be Ramsey distinguishing are the clique number ω and the
odd girth g , where ω is the largest number of vertices in a clique of the graph, and g is
o o
the length of its shortest odd cycle. Specifically, it is shown in [24, 23] that if ω(H) = ω
and g (H) = g then there are Ramsey graphs G,G (H) such that ω(G) = ω and
o o (cid:48)
∈ R
g (G)=g . Note that if Question 1 has a negative answer, then any graph parameter is a
o (cid:48) o
Ramsey distinguishing parameter for the class of connected graphs.
In this paper, we provide a supporting evidence for a ’No’-answer to Question 1 by the
following theorems, focusing on another graph parameter, the chromatic number, χ.
Observation 2. If G and H are graphs, χ(G)=2 and χ(H)>2 then G R H.
(cid:54)∼
Indeed, a sufficiently large complete bipartite graph arrows any fixed bipartite graph [5].
But it does not contain, and thus does not arrow any non-bipartite graph. Here, we prove
that for several large classes of connected graphs, the chromatic number is a Ramsey dis-
tinguishing parameter. A graph is called clique-splittable if its vertex set can be partitioned
into two subsets, each inducing a subgraph of smaller clique number. Note that any graph
G with χ(G) 2ω(G) 2 is clique-splittable. In particular all cliques and all planar graphs
≤ −
containing a triangle are clique-splittable. The triangle-free clique-splittable graphs are
precisely the bipartite graphs.
Theorem 3. If G,H are graphs, G is clique-splittable and χ(G)<χ(H), then G R H.
(cid:54)∼
Corollary 4. If G,H are graphs, χ(G) 2ω(G) 2 and χ(G)=χ(H), then G R H.
≤ − (cid:54) (cid:54)∼
Theorem 3 distinguishes pairs of graphs of distinct chromatic number under some splitta-
bility condition. The following theorem requires stronger assumptions but also applies to
graphsofthesamechromaticnumber. IfH isasubgraph(propersubgraph)ofG, wewrite
H G (H (cid:40)G).
⊆
Theorem 5. Let a connected graph G satisfy the following two properties:
1) There is an independent set S V(G) such that ω(G S)<ω(G).
⊂ −
2) There is a proper χ(G)-vertex-coloring of G in which some two color classes induce a
subgraph of a matching.
Let H be a connected graph, not isomorphic to G, such that either H (cid:40)G or χ(H) χ(G).
≥
Then G R H.
(cid:54)∼
In Theorems 3 and 5 we distinguish pairs of graphs under certain properties. Call a graph
G Ramsey isolated if G R H for any connected graph H not isomorphic to G. Note that
(cid:54)∼
Question 1 asks whether every connected graph is Ramsey isolated or not. We apply the
2
previous results to identify large families of Ramsey isolated graphs. The k-wheel is the
graph on k+1 vertices obtained from a cycle of length k by adding a vertex adjacent to all
vertices of the cycle.
Theorem 6.
1. If G is connected, χ(G) = ω(G) and there is a proper χ(G)-vertex-coloring of G in
which some two color classes induce a subgraph of a matching in G, then G is Ramsey
isolated.
2. Any path and any star are Ramsey isolated.
3. Each connected graph on at most 5 vertices is Ramsey isolated.
Remark 7. If F distinguishes G and H then F has at least min R(G),R(H) vertices.
{ }
The distinguishing graphs used in the proof of Theorem 6 are rather large, except for stars.
However, for most pairs G,H of distinct connected graphs on at most 5 vertices there is a
distinguishing graph on min R(G),R(H) vertices.
{ }
A tree T on k vertices is called balanced if deleting some edge splits T into components
of order at most k+1 each. The extremal function ex(n,H) is the largest number of
(cid:100) 2 (cid:101)
edges in an n-vertex graph with no copy of H. The Erd˝os-S´os-Conjecture states that
ex(n,T) k 2n for any tree T on k vertices. We remark that recently, Ajtai, Koml´os,
≤ −2
Simonovits and Szemer´edi announced a proof of the conjecture for large k [1, 2, 3]. We
state here a much weaker conjecture:
Conjecture 8. There is a positive (cid:15) and an integer n such that ex(n,T) k 1 (cid:15)n for any
(cid:15) ≤ −2−
tree on k vertices and n>n .
(cid:15)
Theorem 9. If Conjecture 8 is true then any two trees of different order are not Ramsey
equivalent. If T is a balanced tree on k vertices and T is any tree on (cid:96) k+1 vertices,
k (cid:96)
≥
then T R T .
k (cid:96)
(cid:54)∼
The next theorem makes use of multicolor Ramsey numbers. Here R(G ,G ,G ) is the
1 2 3
minimum integer n such that any coloring of the edges of K in red, blue, and green has
n
R
a red G , a blue G , or a green G . We write G H if G and H are not Ramsey
1 2 3 k
(cid:54)∼
equivalentink colors,i.e.,thereisagraphΓsuchthatanyk-coloringofitsedgescontainsa
monochromatic G (we write Γ G) and there is such a coloring avoiding monochromatic
k
→
H, or vice versa.
Theorem 10. If G and H are graphs then G R H if one of the following conditions holds:
(cid:54)∼
There is a graph F such that R(G,G,F)<R(H,H,F).
•
G H and there is k 2 with G R H.
k
• ⊆ ≥ (cid:54)∼
The paper is structured as follows. In Section 2 we provide a more detailed summary
of known results on Ramsey equivalence, as well as some observations. In Section 3 we
give known and our preliminary results that will be used in proving the main theorems.
Section 4 contains the proofs of the main results, the Appendix provides lemmas used to
proveRemark7andotherroutinelemmas. Finally,Section5containsconclusionsandopen
questions. We refer the reader to [29] for all standard notations in graph theory and to
Figure 8 for notations used for small graphs. We omit floors and ceilings as long as the
meaning is clear from context. We also assume that the graphs under consideration have at
least one edge.
3
2 Overview and Observations
As mentioned in the introduction, two connected non-isomorphic graphs G and H are not
Ramsey equivalent if one of them is a clique [11], or if ω(G)=ω(H) [24] or when g (G)=
o
(cid:54) (cid:54)
g (H) [23]. Note that it is an open question whether max girth(F) : F G = girth(G)
o
{ → }
for all graphs G [21], where girth(G) is the length of a shortest cycle in G. There are
several other results about Ramsey classes that are useful in checking whether some two
given graphs are Ramsey equivalent or not. Note that two graphs are Ramsey equivalent
if and only if they have the same set of minimal Ramsey graphs. A graph Γ is minimal
Ramsey for H if Γ H, but Γ H for any proper subgraph Γ of Γ. We need to define a
(cid:48) (cid:48)
→ (cid:54)→
few graphs to state the known results: P , C is the path, cycle on n vertices, respectively,
n n
H is a graph on t+1 vertices such that one vertex has degree d and the other vertices
t,d
induce K , K is the complete bipartite graph with parts of sizes a and b, respectively. It
t a,b
was shown in [25] that if H does not contain K , a,b 3 or 1 = a b 2, then there is
a,b
≥ ≤ ≤
Γ (H) such that Γ does not contain K . In general, for an integral graph parameter
a,b
∈ R
s, we define
=min s(Γ):Γ is minimal Ramsey for H .
s
R { }
R
If (H) = (G) then clearly G H. When s = δ, the minimum degree, a number of
s s
R (cid:54) R (cid:54)∼
resultshavebeenobtained: (K )=(t 1)2,[8,12], (H )=t 1,[11], (H )=d2,
δ t δ t,1 δ t,d
R − R − R
for 2 d t, [14], (K )=2min a,b 1, [12], (H)=1 if H is a tree, [28], or when
δ a,b δ
≤ ≤ R { }− R
H isK plusapendingedge,[11], (C )=3forevenn 4,[28]. Burr,Erd˝os,Lov´asz[8]
t,t δ n
R ≥
conjectured that for every integer χ there is H with χ(H)=χ and R (H)=(χ 1)2+1.
χ
−
This conjecture has been proven by Zhu [30]. Burr et al. proved that R (G) R(G) [8].
χ
≤
In[18]theauthorsaskedwhether (H)isboundedbyafunctiondependingonmaximum
∆
R
degree∆(H)only. TheyprovethatifH isatree,then2∆(H) 1 (H) 4(∆(H) 1).
∆
− ≤R ≤ −
Weobservethatforanon-bipartiteH, (H) 2∆(H),seeLemma17. In[26]athreshold
∆
R ≥
t is given, such that almost all graphs with density larger than t are Ramsey for H and
H H
almost all with smaller density are not. The introduction of [6] covers several results on
graphs with infinitely many minimal Ramsey graphs.
It remains unclear whether a connected graph could or could not be Ramsey equivalent to
its subgraph. Note that an edge-transitive graph H is not Ramsey equivalent to any of its
subgraphs because coloring the edges of Γ e for a Ramsey minimal graph Γ of H in two
−
colors, gives a monochromatic copy of H e for some edge e. Since H e is isomorphic
(cid:48) (cid:48) (cid:48)
− −
to H e for any two edges, H e contains any proper subgraph H of H. We see that
(cid:48)(cid:48) (cid:48) (cid:48)
− −
Γ e H , but Γ e H.
(cid:48)
− → − (cid:54)→
3 Preliminary Lemmas
The following lemma is an easy generalization of the Focusing Lemma in [11].
Lemma 11 (Focusing Lemma, [11]). Let (A B,E) be a bipartite graph with a 2-edge-
∪
coloring. Then there is a subset B B, B B /2A, such that for each a A all edges
(cid:48) (cid:48) | |
⊆ | |≥| | ∈
from a to B are of the same color.
(cid:48)
Lemma 12 ([24]). For any graph G there is a graph F (G) with ω(G)=ω(F).
∈R
We write V( ) and E( ) for the vertex set, respectively the edge set, of a graph or hy-
H H
pergraph . A hypergraph is k-uniform if every hyperedge has size k. The girth of a
H
4
hypergraph is the smallest number k 2 of distinct vertices v ,...,v and distinct hy-
0 k 1
≥ −
peredges E ,...,E such that for each i, 0 i k 1, v ,v E (indices taken
0 k 1 i i+1 i
− ≤ ≤ − { } ⊆
modulo k). The independence number of a hypergraph is the size ofa largest set ofvertices
which does not contain a hyperedge completely. The chromatic number of a hypergraph is
thesmallestnumberofcolorsinapropervertexcoloring,i.e.,acoloringwithoutmonochro-
matic hyperedges.
Lemma 13 ([10]). For any integers k,g 2 and any (cid:15) > 0 there is an integer n and a
≥
k-uniform hypergraph on n vertices with girth at least g and independence number less than
(cid:15)n.
From this lemma one easily derives the following well-known result.
Lemma 14 ([10]). For any integers k,g,χ 2 there is a k-uniform hypergraph with girth
≥
at least g and chromatic number at least χ.
Proof. Let (cid:15) < 1 and let denote a k-uniform hypergraph with girth at least g and
χ H
independence number at most (cid:15)V( ), which exists by Lemma 13. In any χ-coloring of
| H |
V( ) there is a set of at least 1 V( ) > (cid:15)V( ) vertices of the same color. Hence this
H χ| H | | H |
color class induces an edge of . Thus the coloring is not proper and χ( )>χ.
H H
(cid:15)
For graphs F, G and for (cid:15)>0 we write F G if for any set S V(F) with S (cid:15)V(F),
→ ⊆ | |≥ | |
we have F[S] G. Here F[S] denotes the subgraph of F induced by the vertices in S.
→
Lemma 15 ([11]). For any (cid:15)>0 and any graph H, there is a graph F with ω(F)=ω(H)
and F (cid:15) H.
→
Proof. Let F be a graph such that F H and ω(F ) = ω(H). Such a graph exists by
(cid:48) (cid:48) (cid:48)
→
Lemma 12. Further let denote a V(F )-uniform hypergraph of girth at least 4 and no
(cid:48)
H | |
independentsetofsize(cid:15)V( ),whichexistsbyLemma13. ConstructagraphF byplacing
| H |
a copy of F on the vertices of each hyperedge of . Then F is a graph on V( ) vertices
(cid:48)
H | H |
with ω(F) = ω(F ) = ω(H). Each vertex set of size at least (cid:15)V( ) induces a hyperedge
(cid:48)
| H |
in and thus a copy of F in F which arrows H.
(cid:48)
H
We have the following corollary, since any graph which arrows H contains H.
Lemma 16. For any (cid:15) > 0 and any graph H, there is a graph F with ω(F) = ω(H) and
each set of (cid:15)V(F) vertices in F containing a copy of H.
| |
Lemma 17. If a graph H is not bipartite then (H) 2∆(H). The lower bound is tight.
∆
R ≥
Proof. Let ∆ = ∆(H) and suppose F is a graph with ∆(F) 2∆ 1. It is sufficient to
≤ −
prove that F H. Consider a partition V ˙V of V(F) with the maximum number of
1 2
(cid:54)→ ∪
edges between V and V . If there is a vertex v V with at least ∆ neighbors in V , then
1 2 1 1
∈
v has at most ∆ 1 neighbors in V . Thus the partition (V v )˙(V v ) has at least
2 1 2
− \{ } ∪ ∪{ }
one more edge between the parts than the original partition, a contradiction. Hence both
F[V ] and F[V ] have maximum degree at most ∆ 1. Color all edges between V and V
1 2 1 2
−
red and all other edges blue. Then the red subgraph is bipartite and the blue subgraph has
maximum degree at most ∆ 1. Thus F H.
− (cid:54)→
The lower bound is tight since K H, ∆ 3, where H is the graph of maximum
2∆+1
→ ≥
degree ∆ obtained from K by adding an edge between two leaves.
1,∆
5
Lemma 18. Let G and H be graphs.
If ex(n,G)<ex(n,H)/2 or if H is connected and ex(n,G)<√nex(√n,H), then G R H.
(cid:54)∼
In particular, if G is a forest and H contains a cycle, then G R H.
(cid:54)∼
Proof. Assume first that ex(n,G) < ex(n,H)/2. Let F be a graph on n vertices with
ex(n,H) 2ex(n,G)+1 edges without a copy of H. In any 2-coloring of the edges of F
≥
one of the color classes contains at least ex(n,G)+1 edges, and thus a copy of G. Hence
F G, but F H.
→ (cid:54)→
AssumenowthatH isconnectedandex(n2,G)<nex(n,H). LetF beagraphonnvertices
and ex(n,H) edges not containing H. Let F =F F be the Cartesian product of F with
∗
×
itself, that is, V(F ) = V(F) V(F) and (u,v),(x,y) E(F ) if and only if u = x and
∗ ∗
× { } ∈
vy E(F) or v = y and ux E(F). Then F has n2 vertices and 2nex(n,H) edges. In
∗
∈ ∈
any 2-edge-coloring of F there is a color class with at least nex(n,H) edges. This color
∗
class contains a copy of G, thus F G. On the other hand, we can color the edges of
∗
→
E(F ) without creating monochromatic copies of H by coloring an edge (u,v),(x,y) red
∗
{ }
if u=x and blue otherwise. Note that each color class is a vertex disjoint union of n copies
of F and thus does not contain H, as H is connected. Thus F H.
∗
(cid:54)→
For the second part of the statement let G be any forest and H be any graph with a cycle
C. We have ex(n,G) V(G)n and ex(n,H) ex(n,C) Ω(n1+|V(C1)|−1) [20]. Hence for
≤ | | ≥ ≥
R
sufficiently large n we have ex(n,G) < ex(n,C)/2 and thus G H by the first part of the
(cid:54)∼
Lemma.
Let Z and Z denote the graphs obtained from C by adding two, respectively three,
1 4 4
pendent edges at some vertex, let Z denote the graph obtained from K by adding a
5 2,3
pendent edge at a vertex of degree 3, see Figure 1.
Z Z Z Z
1 2 3 4
Z
5
Figure 1: The graphs Z , Z , Z , Z and Z .
1 2 3 4 5
The next technical lemmas we prove in the Appendix.
Lemma 19. (Z )=1.
δ 4
R
Lemma 20. In any 2-edge-coloring of K e without monochromatic Z the vertex of
3,13 5
−
degree 2 is incident to exactly one red and one blue edge.
Lemma 21. (Z )=1.
δ 5
R
Lemma 22. If H is connected and has at least 6 vertices, then H R W .
4
(cid:54)∼
Lemma 23. If H is connected and has at least 6 vertices, then H R K .
2,3
(cid:54)∼
6
4 Proofs of Theorems
4.1 Proof of Theorem 3
If χ(G) = 2 then G is not Ramsey equivalent to any graph H of higher chromatic number
by Observation 2. So, assume that χ(G) 3.
≥
R
Let ω = ω(G), k = χ(G), χ(H) > k. We assume ω(G) = ω(H), otherwise G H by
(cid:54)∼
Lemma 12. Let V(G) = V V , V V = , such that G = G[V ] and G = G[V ]
1 2 1 2 1 1 2 2
∪ ∩ ∅
each have clique number less than ω. Let G(cid:63) be a vertex disjoint union of G and G , in
1 2
particular ω(G(cid:63))<ω. We shall construct a graph Γ such that Γ G and Γ H.
→ (cid:54)→
The building blocks of Γ are a hypergraph and graphs F and F such that:
(cid:48)
H
is a 3-chromatic, k-uniform hypergraph of girth at least V(H) +1. It exists by
• H | |
Lemma 14.
F is a graph such that ω(F) < ω and every set of at least (cid:15) V(F) vertices in F
1
• | |
contains a copy of G(cid:63), where (cid:15) =2 V( ). Such a graph exists by Lemma 16.
1 −| H |
F is a graph such that ω(F )=ω(F)<ω and F (cid:15) F for (cid:15)=2 V( ) V(F). Such a
(cid:48) (cid:48) (cid:48) −| H || |
• →
graph exists by Lemma 15.
Note that V( ) depends on V(H) and χ(G); V(F) in turn depends on V( ), ω(G),
| H | | | | | | H |
and G, so V(F) depends only on H and G. So, (cid:15) and (cid:15) are constants depending on H
1
| |
and G.
Construct a graph Γ by replacing the vertices v ,...,v of with pairwise vertex disjoint
1 n
H
copies of F on vertex sets V ,...,V and placing a complete bipartite graph between two
(cid:48) 1 n
copies of F if and only if the corresponding vertices belong to the same hyperedge of ,
(cid:48)
H
see Figure 2.
F0 F0
F
0
F Γ
H 0
F
0
F0 F0
F
0 F
0
F0 F0
F
0
Figure 2: Left: The k-uniform hypergraph , k =3. Right: The graph Γ.
H
To show that Γ H color each edge with both endpoints in some V red, i = 1,...,n,
i
(cid:54)→
and all other edges blue. The red subgraph is a vertex disjoint union of copies of F , its
(cid:48)
clique number is strictly less than ω, so it does not contain H, whose clique number is ω.
The blue subgraph is a union of complete k-partite graphs induced by V , i = 1,...,n. To
i
see that the blue subgraph does not contain a copy of H, consider any copy of H in Γ and
consider sets V ,...,V intersecting the vertex set of this copy. Since has girth at least
i1 i(cid:96) H
V(H) +1,v ,...,v donotformacyclein ,thusthebluegraphinducedbyV ,...,V
| | i1 i(cid:96) H i1 i(cid:96)
7
is k-partite. However, χ(H)>k, so the blue subgraph does not contain a copy of H.
NextweshallshowthatΓ G. Considera2-edge-coloringofΓ. Recallthatn= V( ) =
→ | H |
n(k, V(H)). We write v v if there is a hyperedge in containing both v and v .
i j i j
| | ∼ H
Claim 1. For any m, 1 m n, any i, 1 i m, V contains a subset V that is
≤ ≤ ≤ ≤ i i(cid:48)
the vertex set of a monochromatic copy of F and such that for any v V and any j with
∈ i(cid:48)
v v , i<j m, all edges from v to V , are of the same color.
i ∼ j ≤ j(cid:48)
V V V V V V V V
10 20 30 ··· m0 100 200 300 ··· m00
Figure 3: Illustrations of Claim 1 (left) and Claim 2 (right). Thin edges are red and thick
edges blue.
WeproveClaim1byinductiononmusingtheFocusingLemma(Lemma11). Whenm=1,
(cid:15)
we see that Γ[V ] is isomorphic to F and F F. So in particular F F and there is a
1 (cid:48) (cid:48) (cid:48)
→ →
monochromatic copy of F on some vertex set V . Assume that V ,V ,...,V form vertex
1(cid:48) 1(cid:48) 2(cid:48) m(cid:48)
sets ofmonochromatic copies of F satisfying theconditions ofClaim 1. Apply theFocusing
Lemma to the bipartite graph with parts U = V V and V . It gives a subset
m 1(cid:48)∪···∪ m(cid:48) m+1
V V such that for any v U , all edges between v and V , if any, are of the
m∗+1 ⊆ m+1 ∈ m m∗+1
samecolorandsuchthat|Vm∗+1|≥2−|V1(cid:48)∪···∪Vm(cid:48)||Vm+1|=2−m|V(F)||Vm+1|≥(cid:15)|Vm+1|. Thus
Γ[V ] contains a monochromatic copy of F, because Γ[V ] is isomorphic to F and
m∗+1 m+1 (cid:48)
(cid:15)
F F. Call the vertex set of this copy V .
(cid:48) → m(cid:48)+1
Claim 2. For any m, 1 m n, and i, m i n, each V contains a subset V V
≤ ≤ ≤ ≤ i i(cid:48)(cid:48) ⊆ i(cid:48)
that is the vertex set of a monochromatic copy of G(cid:63) and such that for each j with v v ,
i j
∼
i<j n, V , V are partite sets of a monochromatic complete bipartite graph.
≤ i(cid:48)(cid:48) j(cid:48)(cid:48)
WeproveClaim2byinductiononn musingthepigeonholeprinciple. Whenm=n,wesee
−
that V forms the vertex set of a monochromatic F, that in turn contains a monochromatic
n(cid:48)
G(cid:63). Denote the vertex set of this G(cid:63) as V . Assume that V ,V ,...,V form vertex
n(cid:48)(cid:48) m(cid:48)(cid:48) m(cid:48)(cid:48)+1 n(cid:48)(cid:48)
setsofmonochromaticcopiesofG(cid:63) satisfyingtheconditionsofClaim2. ConsiderV and
m(cid:48) 1
recall from Claim 1 that each vertex in V sends only red or only blue edges to ea−ch V
m(cid:48) 1 i(cid:48)(cid:48)
with v v , i=m,...,n. If v v−then at least half of the vertices in V send
monocmhr−o1m∼atici stars of the same mco−lo1r∼tonV . If v v then at least halfm(cid:48)o−f1those
send monochromatic stars of the same color nt(cid:48)o(cid:48) V m,−an1d∼sono−n1. So at least 2 (n m) V
vertices of V send monochromatic stars of thn(cid:48)e(cid:48)−1same color to each V wi−th −v | m(cid:48)−v1|
for i = m,..m(cid:48).−,n1. We denote the set of these vertices by V . Since Γi(cid:48)(cid:48)[V ] fmor−m1s∼thei
m∗ 1 m(cid:48) 1
vertex set of a monochromatic F, and V 2 (n m) V − (cid:15) V ,−the definition
of F implies that Γ[V ] contains a m|onm∗o−ch1r|o≥ma−tic −copy| om(cid:48)f−G1(cid:63)|.≥We1|dem(cid:48)n−o1t|e the vertex set
m∗ 1
of this copy by V . −
m(cid:48)(cid:48) 1
Applying Claim 2 w−ith m=1, we see that each vertex v of corresponds to a monochro-
i
H
matic copy of G(cid:63) with vertex set V , such that all edges between any two such copies from
i(cid:48)(cid:48)
8
G?
G?
G?
Figure 4: A set of k red copies of G(cid:63) corresponding to the k vertices of a hyperedge of
. Here k = 3. The complete bipartite graph between any two of these k copies is also
H
monochromatic.
a common hyperedge have the same color. Assigning the color of this G(cid:63) to v gives a
i
2-coloring of V( ). Since χ( ) > 2, there is a monochromatic hyperedge, without loss of
H H
generalitywithredverticesv ,...,v . ThusinΓtherearek redcopiesofG(cid:63) onvertexsets
1 k
V ,...,V , such that V ,V are partite sets of monochromatic complete bipartite graphs,
1(cid:48)(cid:48) k(cid:48)(cid:48) i(cid:48)(cid:48) j(cid:48)(cid:48)
foralli,j, 1 i<j k, seeFigure4. Ifatleastonesuchbipartitegraphisred,thenthere
≤ ≤
is a red copy of G obtained by taking a red G G from one part and a red G G
1 ∗ 2 ∗
⊆ ⊆
from the other part. So we can assume that all such bipartite graphs are blue, forming a
complete k-partite graph with each part of size V(G). Since χ(G) = k, there is a blue
| |
R
copy of G. Thus Γ G. Since Γ H, we have that G H. This concludes the proof of
→ (cid:54)→ (cid:54)∼
Theorem 3.
Proof of Corollary 4. Let G and H be two graphs such that χ(G) = χ(H) and χ(G)
(cid:54) ≤
2ω(G) 2. Consider an arbitrary proper χ(G)-vertex-coloring of G. Let V denote the
1
−
union of χ(G) color classes and V =V(G) V . Since ω(G) χ(G) +1, every maximum
(cid:98) 2 (cid:99) 2 \ 1 ≥ 2
clique contains a vertex from both sets V , i = 1,2. Thus, G is clique splittable. So, if
i
R
χ(H)>χ(G) then G H by Theorem 3. If χ(H)<χ(G), then χ(H)<χ(G) 2ω(H) 2
(cid:54)∼ ≤ −
(where we assume ω(H)=ω(G) by Lemma 12). Thus, H is clique-splittable with the same
R
arguments as above. Hence G H by Theorem 3.
(cid:54)∼
4.2 Proof of Theorem 5
Our construction is similar to the one from Lemma 3.9 in [14].
Consider a connected graph H. We may assume ω(G)=ω(H) by Lemma 12. Note that G
isclique-splittablesinceω(S)=0andω(G S)<ω(G). FurthernotethatifGisbipartite,
−
theconditionsofthetheoremimplythatGisaunionofamatchingandasetofindependent
vertices. However, G is assumed to be connected, and thus it must be a single edge. Since
a single edge is Ramsey isolated, we can assume that χ(G) 3.
≥
In the first part of the proof, we assume that H G and χ(H) χ(G). Let s = S ,
(cid:54)⊆ ≥ | |
k = χ(G) = χ(H) and let m denote the size of a matching induced by two color classes
of some proper k-vertex-coloring of G. Note that m 1 since there is at least one edge
≥
between any two color classes. Further let n = V(G), ω = ω(G) and G be a vertex
S
| |
disjoint union of G S and S independent vertices, i.e., G is the graph obtained from G
S
−
9
by deleting all edges incident to S. Then ω(G ) < ω. Let G be a vertex disjoint union of
S (cid:48)
m =(k 2)(s 1)+m copies of G and G be a vertex disjoint union of m copies of G.
Le(cid:48)t (cid:15)=2− V(G(cid:48)−)k =2 m(cid:48)nk. Let F bSe a grap(cid:48)0h with F (cid:15) G and ω(F)=ω(G(cid:48))<ω, which
−| | − (cid:48) (cid:48)
→
exists by Lemma 15. We construct a graph Γ by taking the vertex disjoint union of a copy
ofG andk 2copiesofF denotedbyF ,...,F andplacingacompletebipartitegraph
(cid:48)0 − 1 k−2
betweenF andF , 1 i<j k 2andbetweenF andG , i=1,...,k 2, seeFigure5.
i j ≤ ≤ − i (cid:48)0 −
G
00
···
··· G G
S
F1 ···
G =m copies of G
0 0 S
···
··· ···
F2 F G00 =m0 copies of G
··· k−2
Figure 5: The graph Γ consisting of k 2 copies F ,...,F of F and one copy of G and
− 1 (cid:15) k−2 (cid:48)0
all possible edges between distinct copies. We have F G, G consists of m copies of G
(cid:48) (cid:48) (cid:48) S
→
and G consists of m copies of G.
(cid:48)0 (cid:48)
We shall show that Γ G, but Γ H. Color all edges within each F and within G red
→ (cid:54)→ i (cid:48)0
andallotheredgesblue. Sinceω(F)<ω =ω(H),H F,andthusH F ,i=1,...,k 2.
i
(cid:54)⊆ (cid:54)⊆ −
SinceH GandH isconnected,wehavethatH G . ThusthereisnoredcopyofH. On
(cid:54)⊆ (cid:54)⊆ (cid:48)0
theotherhand,thebluesubgraphisacomplete(k 1)-partitegraph,butχ(H) χ(G)=k.
− ≥
Thus there is no blue copy of H.
It remains to show that Γ G. Consider a 2-edge-coloring of Γ. Assume for the sake
→
of contradiction that there is no monochromatic copy of G. We prove the following claim,
similar to Claim 1 in the proof of Theorem 3, by induction on p (up to renaming colors),
see Figure 6 for an illustration.
Claim. For each p, 1 p k 2, and each i, 1 i p, there is a red copy G of G in
≤ ≤ − ≤ ≤ (cid:48)i (cid:48)
F . Moreover for each i, 0 i < p, each vertex v in G and each j, i < j p, all edges
i ≤ (cid:48)i ≤
between v and G are of the same color.
(cid:48)j
There is a set V of 2 m(cid:48)n V(F ) (cid:15)V(F) vertices in F such that for each vertex in G
1 − | 1 |≥ | | 1 (cid:15) (cid:48)0
all edges to V are of the same color by the Focusing Lemma (Lemma 11). Since F G
1 (cid:48)
→
there is a monochromatic copy G of G in F [V ]. Assume without loss of generality that
(cid:48)1 (cid:48) 1 1
G is red. This proves the Claim for p=1, and for k =3.
(cid:48)1
Suppose k 2 p 2 and there are red subgraphs G ,...,G , satisfying the conditions
of the Clai−m. ≥We≥apply the Focusing Lemma to the(cid:48)1 comple(cid:48)pt−e1bipartite graph with one
part V(G ) V(G ) and the other part V(F ). There is a set V V(F ) of size
(cid:48)0 ∪···∪ (cid:48)p−1 p p ⊆ p
10
