Edge Colorings of
Complete Graphs Without
Tricolored Triangles
Andra
´
s Gya
´
rfa
´
s
1
and Ga
´
bor Simonyi
2
1
COMPUTER AND AUTOMATION RESEARCH INSTITUTE
OF THE HUNGARIAN ACADEMY OF SCIENCES
BUDAPEST, P.O. BOX 63, 1518, HUNGARY
E-mail: gyarfas@sztaki.hu
2
ALFRED RE
´
NYI INSTITUTE OF MATHEMATICS
HUNGARIAN ACADEMY OF SCIENCES
BUDAPEST, P.O. BOX 127, 1364, HUNGARY
E-mail: simony@renyi.hu
Received June 3, 2002; Revised November 4, 2003
DOI 10.1002/jgt.20001
Abstract: We show some consequences of results of Gallai concerning
edge colorings of complete graphs that contain no tricolored triangles. We
prove two conjectures of Bialostocki and Voxman about the existence
of special monochromatic spanning trees in such colorings. We also deter-
mine the size of largest monochromatic stars guaranteed to occur.
ß 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004
Keywords: Gallai coloring; monochromatic spanning tree; Ramsey theory
——————————————————
Contract grant sponsor: OTKA (to A.G.); Contract grant number: T029074;
Contract grant sponsor: OTKA (to G.S.); Contract grant numbers: T032323,
T037486.
ß 2004 Wiley Periodicals, Inc.
211
1. INTRODUCTION
We consider edge colorings of complete graphs in which no triangle is colored
with three distinct colors. These colorings generalize 2-colorings and we shall
call them Gallai colorings. A similar terminology, Gallai partition, is used in
Ref. [10]. The reason is the close connection of these colorings to the basic work
[8] of Gallai on comparability graphs. Gallai colorings also appear in Ref. [5], a
paper of Cameron, Edmonds and Lova
´
sz where the (weak) perfect graph theorem
[11] is extended (see Theorem C below). They turned out to be relevant also in
investigations concerning the additivity properties of the information theoretic
functional called graph entropy (see Refs. [9,10]).
In this paper we look at some Ramsey-type problems for Gallai colorings. The
first problems of this type were studied by Erdo
´´
s, Simonovits and So
´
s in Ref. [7]
where it was shown that Gallai colorings of K
n
can use at most n 1 colors (see
Proposition B). Equality holds for the coloring where, for i ¼ 1; 2; ...; n 1
color i forms a star of i edges. This shows that a Gallai coloring may be such that
all of its monochromatic subgraphs are stars. Gallai colorings, like (their special
case of) 2-colorings, always have monochromatic spanning trees. This is observ-
ed by Bialostocki, Dierker and Voxman in [1]. In Ref. [2] (also in Ref. [1])
Bialostocki and Voxman raise three problems about the existence of specific
monochromatic spanning trees in Gallai colorings. We answer these problems
(two of them positively) as follows. In any Gallai coloring, there is a mono-
chromatic spanning broom, where a broom is a path with a star at its end
(Theorem 2.1). Burr (in Ref. [3]) proved this for 2-colorings (conjectured also
by Bialostocki). Gallai colorings also contain monochromatic spanning trees of
height two (Theorem 2.2). We also prove that the largest monochromatic star
which must appear in any Gallai coloring of K
n
has at least 2n=5 edges
(Theorem 3.1). An easy construction shows that this bound is sharp, implying a
negative answer to the third question of Bialostocki and Voxman.
It is obvious that Gallai colorings are closed under substitution: replacing a
vertex in a Gallai coloring by a complete graph with a Gallai coloring gives again
a Gallai coloring. The following important result shows that all Gallai colorings
can be obtained by substituting into 2-colored complete graphs. Theorem A (and
Lemma A) is implicit in Ref. [8] and also among the results of Cameron and
Edmonds on Lambda composition (see Ref. [4]). Due to its importance (and to
keep the paper self-contained) we state and prove it.
Theorem A. Any Gallai coloring can be obtained by substituting complete
graphs with Gallai colorings into vertices of 2-colored complete graphs.
Theorem A can be applied to extend results from 2-colorings to Gallai
colorings. We shall refer to the 2-colored complete graph as the base graph and
the graphs substituted into the vertices of the base graph will be called the blocks.
Theorem A will be derived from the following property of Gallai colorings
which is essentially the content of Lemma (3.2.3) in Ref. [8].
212 JOURNAL OF GRAPH THEORY
Lemma A. Every Gallai coloring with at least three colors has a color which
spans a disconnected graph.
Now Theorem A is obvious from Lemma A: If a Gallai coloring is just a
2-coloring, we are done. Otherwise we have a color with at least two components.
It is clear that edges between any two components are colored with the same
color. Collapsing the components into vertices, we have a smaller graph with a
Gallai coloring which, by induction, can be generated as required.
Proof. Let G be a minimal counterexample, clearly all colors appear on some
edge incident to any particular vertex of G. Let x 2 VðGÞ and H ¼ Gnx. Then H
cannot be 2-colored because then any other color would span a disconnected
graph. By minimality, H is disconnected in some color, say in color 1 with
components C
1
; ...; C
k
. As noted before, all edges between any fixed pair of
components have the same color (different from 1).
We claim that G is disconnected in color 1. Indeed, assume that there are edges
of color 1 from x to y
i
2 C
i
for every i. Let xu and xv be edges of color 2 and 3.
Case 1. If u; v are in the same component, say u; v 2 C
1
then uy
2
must be of
color 2 and vy
2
must be of color 3 (using that the triangles xuy
2
and xvy
2
are not
tricolored). This contradicts the homogeneous coloring of the edges between
C
1
and C
2
.
Case 2. If u; v are in different component, say u 2 C
1
; v 2 C
2
then uy
2
must be
of color 2 and v y
1
must be of color 3 (using that the triangles xuy
2
and xvy
1
are
not tricolored). We get the same contradiction as in Case 1.
Therefore, the claim is proved, G is disconnected in color 1. Thus G cannot be
a counterexample.
&
Theorem A can be conveniently used to derive properties of Gallai colorings.
The following result is from Ref. [7].
Proposition B. At most n 1 colors can be used in any Gallai coloring of K
n
.
Proof. Apply induction for the blocks of the base graph.
&
Theorem A can also be used to give the following generalization of the
(weak) perfect graph theorem. The theorem is from Ref. [5], its relation to
Gallai’s work is further emphasized in Ref. [4]. (See Ref. [10] for a generalization
of Theorem C where Lemma A also plays an important role.)
Theorem C. If all but one color classes of a Gallai coloring span perfect
graphs then all color classes span perfect graphs.
2. MONOCHROMATIC SPANNING TREES
IN GALLAI COLORINGS
An old remark of Paul Erdo
´´
s says that 2-colored complete graphs have mono-
chromatic spanning trees. One can also say something about the type of the
EDGE COLORING WITHOUT TRICOLORED TRIANGLES 213
spanning tree. Bialostocki, Dierker, and Voxman proved in Ref. [1] that there is a
monochromatic spanning tree of height at most two. Burr [3] proved, answering
the conjecture of Bialostocki, that there is a spanning ‘broom,’ which means the
union of a path and a star with the central vertex of the latter identified with an
endvertex of the former. Bialostocki and Voxman conjectured (Ref. [2], Problems
3.3a and c) that both results can be generalized to Gallai colorings. Theorems 2.1
and 2.2 verify these conjectures using Theorem A. The essential steps in the proof
of Theorem 2.1 follow Burr’s nice (unpublished) proof [3].
Theorem 2.1. In every Gallai coloring of a complete graph K there is a mono-
chromatic spanning broom.
Proof. Using Theorem A, we assume that the base graph of K is colored
with colors red and blue. Without loss of generality, assume that the red edges
determine a k-connected graph and the blue edges determine an at most k-
connected graph on the vertex set of K (k is a positive integer). This implies that
there is a subset A with jAjk whose removal disconnects the blue graph. We
may also assume that jAj is as small as possible, i.e., A is a minimal separator of
the blue graph. If A is empty, i.e., the blue graph is disconnected, then the vertices
of K are spanned by a red complete bipartite graph which obviously contains
a red spanning broom. Therefore, A is nonempty. By definition, VðKÞnA has a
nontrivial partition into X; Y such that there are no blue edges between X and Y.
Claim 2.1. X [ Y has a red spanning complete bipartite graph H.
If there are no vertices x 2 X and y 2 Y such that x; y belong to the same block
of the base graph then all edges between X and Y are red and the claim is proved.
Otherwise there is a block B of the base graph such that U ¼ B \ X; V ¼ B \ Y
are nonempty. It follows that all edges between U [ V and ðX [ YÞnðU [ VÞ are
red—unless U ¼ X and V ¼ Y. However, in this case every vertex of AnB sends a
blue edge to B, thus all edges between B and AnB are blue. This implies that the
base graph is disconnected in red, a contradiction. Thus the claim is proved.
Now the proof is finished by applying a well-known result of Dirac [6] which
says that any k vertices of a k-connected graph can be covered by a cycle of at
least k þ 1 vertices. We use this theorem for the k-connected red graph and the k
vertices in A. Let the cycle guaranteed to exist by Dirac’s theorem be C. (In the
degenerate case when k ¼ 1, C is defined as a red edge containing the vertex of
A.) Thus the vertex set of K is covered by C [ H. Using that C and H have
nonempty intersection one can easily find a red spanning broom.
&
Theorem 2.2. In every Gallai coloring, there is a monochromatic spanning tree
with height at most two.
Proof. By Theorem A, the Gallai coloring can be obtained by substitutions
into a 2-colored base graph H. It is easy to see (cf. Theorem 2.1 in Ref. [1]) that H
has a monochromatic spanning tree T with height at most two (the root can be
any vertex with maximum monochromatic degree). One can easily extend T:
214 JOURNAL OF GRAPH THEORY
substituting a set X into a nonroot vertex x of T results in adding each element of
X as a leaf with the same father; substituting into the root of T results in adding
each element but one (that remains the root) of X as a leaf with its father at an
arbitrary vertex of level one in T.
&
3. MONOCHROMATIC STARS IN GALLAI COLORINGS
It is a natural question to ask for the maximum monochromatic degree in a Gallai
coloring of K
n
. Consider the red-blue coloring of K
5
where both color classes
form pentagons. Substituting green complete graphs into this base graph, one can
get a Gallai colored K
n
with no monochromatic degree exceeding 2n=5. This
construction is best possible as shown by the next theorem (and provides a
negative answer to Problem 3.3b in Ref. [2]).
Theorem 3.1. Any Gallai coloring of K
n
has a color with largest degree at least
2n=5.
Proof. By Theorem A, the Gallai coloring can be defined by substituting into
a base graph colored with colors 1,2. It is easy to check that if the base graph has
at most four vertices then color 1 or 2 has degree at least n=2. If the base graph
has at least five vertices then there is a block B with at most n=5 vertices.
Therefore, any vertex in B is adjacent to at least 4n=5 vertices outside B in colors
1 or 2 and the theorem follows.
One can use Theorem 3.1 in the proof of Theorem 2.2 to show that the root of
the monochromatic spanning tree found there can be of degree at least 2n=5.
REFERENCES
[1] A. Bialostocki, P. Dierker, and W. Voxman, Either a graph or its complement
is connected: A continuing saga (manuscript in preparation).
[2] A. Bialostocki and W. Voxman, On monochromatic-rainbow generalizations
of two Ramsey-type theorems. Ars Combinatoria 68 (2003), 131–142.
[3] S. A. Burr, Either a graph or its complement contains a spanning broom
(manuscript in preparation).
[4] K. Cameron and J. Edmonds, Lambda composition, J Graph Theory 26
(1997), 9–16.
[5] K. Cameron, J. Edmonds, and L. Lova
´
sz, A note on perfect graphs, Period
Math Hungar, 17 (1986), 173–175.
[6] G. A. Dirac, In abstrakten Graphen vorhandene vollsta
¨
ndige 4-graphen und
ihre Unterteilungen, Math Nachr 22 (1960), 61–85.
[7] P. Erdo
´´
s, M. Simonovits, and V. T. So
´
s, Anti-Ramsey theorems, Coll Math
Soc J Bolyai 10 (1973), 633–643.
EDGE COLORING WITHOUT TRICOLORED TRIANGLES 215