Unavoidable chromatic patterns in colorings
of the complete graph
Abstract
Let be a graph with edges. We say that is omnitonal if for every sufficiently large there exists a minimum integer such that the following holds true: For any coloring with more than edges from each color, and for any pair of nonnegative integers and with , there is a copy of in with exactly red edges and blue edges. We give a structural characterization of omnitonal graphs from which we deduce that omnitonal graphs are, in particular, bipartite graphs, and prove further that, for an omnitonal graph , , where depends only on . We also present a class of graphs for which , the celebrated Turán numbers. Many more results and problems of similar flavor are presented.
Yair Caro
[1ex] Dept. of Mathematics
University of HaifaOranim
Tivon 36006, Israel
Adriana Hansberg
[1ex] Instituto de Matemáticas
UNAM Juriquilla
Querétaro, Mexico
[2ex]
Amanda Montejano
[1ex] UMDI, Facultad de Ciencias
UNAM Juriquilla
Querétaro, Mexico
[4ex]
1 Introduction and problem setting
Our main interest in this paper is a certain kind of problems that lie in the junction of Ramsey theory, extremal graph theory, zerosum Ramsey theory and interpolation theorems in graph theory; general references to these topics are [1, 2, 3, 5, 14, 16, 17, 18].
We consider colorings of the set of edges of the complete graph . Given a graph with edges, nonnegative integers and such that , and a coloring , we say that induces an colored copy of , if there is a copy of in such that assigns the color red to exactly edges and the color blue to exactly edges of that copy of .
By Ramsey’s theorem we know that any coloring of (where is sufficiently large) induces either a colored copy of , or a colored copy of . To force the existence of with other color patterns, we need, as a natural minimum requirement, not only to ensure a large , but also a minimum amount of edges of each color. In this paper, we study which graphs are unavoidable under a prescribed color pattern in every coloring of , whenever is sufficiently large and there are enough edges from each color. A similar approach has been studied in [4, 10, 12], where the emphasis is given on determining the minimum required to guarantee the existence of a given graph with a prescribed color pattern in every coloring of where each color appears in some positive fraction of the edges of . In contrast, our approach has more like a Turán flavor in the sense that we focus our attention, on the one hand, on the maximum edge number that can have the smallest color class in a coloring of which is free of a copy of the given graph in the prescribed pattern, and, on the other, in characterizing the extremal colorings.
Observe that, in case (mod ), the study of the existence of a zerosum copy of over weightings of , in particular over weightings of , carries along similarity to classical Ramsey theory by simply defining all red edges to have weight and all blue edges to have weight . Thus, a zerosum copy of translates into a copy of with equal number of red and blue edges, or equivalently to an colored copy of . The study of the existence of such a balanced copy will be one of the purposes of this paper, thus further developing the line of research studied in [6, 8, 9]. The second problem we will focus on, and in which we will make use of interpolation techniques, deals with the existence of an colored copy of for every pair of nonnegative integers and such that . As in the previous case, this problem is related to the study of the existence of a zerosum copy of over weightings of with range , where and are positive integers with and (mod ). These problems will lead us to the definition of two graph families which will be the center of this work: balanceable graphs and omnitonal graphs.
1.1 Notation
For a given graph , we use and to denote the sets of vertices and, respectively, of edges of . Given a partition of the vertex set , we denote by the set of edges of with one end in and the other one in . Also, we define , and . For a set , stands for the subgraph of induced by the vertices in . Similarly, if , denotes the graph induced by the edges from . A set is called independent if is edgeless.
Let , and be nonnegative integers. We will denote with the complete bipartite graph with one partition set having vertices an the other . The graph will be also called a star. Moreover, a path denotes a path on vertices and edges. A graph is a split graph if there is a partition of the vertex set with and such that induces a complete graph and is an independent set. Furthemore, the split graph is called complete if .
A coloring of the edges of a graph is a mapping to a set of colors . If , we talk about a edge coloring, or for short a coloring. In the whole paper, we will deal with red and blue colorings on the complete graph, that is, we will consider mappings . Since such a coloring induces a partition of the edge set of into the set of red edges and the set of blue edges, we can talk about the red graph and the blue graph induced by the red and, respectively, blue edges of this coloring. In the following, in order to avoid talking every time of the mapping, we will talk about the coloring , assuming implicitly that and are the graphs induced by the red and the blue edges.
1.2 Problem setting
Balanceable graphs
For a given graph , we say that a coloring contains a balanced copy of , if, in the so colored , we can find an colored copy of in case (mod ), and a colored copy of or an colored copy of in case (mod ).
Definition 1.1.
For a given graph let be the minimum integer, if it exists, such that any coloring with contains a balanced copy of . If exists for every sufficiently large , we say that is balanceable^{1}^{1}1For graphs with an odd number of edges, there is a stronger notion of balanceable graphs that one can naturally consider: instead of seeking for one of both, a colored copy or a colored copy of , one can study the existence of both patterns.. For a balanceable graph , let be the family of graphs with exactly edges such that a coloring with contains no balanced copy of if and only if or is isomorphic to some .
We shall be interested in finding balanceable graphs as well as in, if possible, determining or, if not, in finding good estimates. If is known, we also consider the problem of characterizing the extremal colorings, meaning that we aim to determine the family . Note that, to prove that a graph is not balanceable, it is enough to exhibit infinitely many values of for which there is a edge coloring of with the same or almost the same (differing by at most one unit) number of red and blue edges without a balanced copy of .
As we shall show, there is a plethora of balanceable graphs, including but not any other complete graph with even, as proved in [8], where and were also determined.
The connection to the zerosum analogue with weightings is already explained in the introduction section. The case when (mod ) has no direct analogue as a zerosum problem. However, we mention that such oddcase variations have been considered in [9] in the context of sequences. This establishes the bridge between the current paper and the results on weightings given for instance in [6, 8, 9].
Omnitonal graphs
As we will formally define below, omnitonal graphs will be those graphs that appear in all possible tonal variations of red and blue in every edge coloring of the complete graph, as long as the latter is large enough.
Definition 1.2.
For a given graph , let be the minimum integer, if it exists, such that any coloring with contains an colored copy of for any and such that . If exists for every sufficiently large , we say that is omnitonal. For an omnitonal graph , let be the family of graphs with exactly edges such that a coloring with contains no colored copy of for some pair with if and only if or is isomorphic to some .
We shall be interested in finding omnitonal graphs as well as in, if possible, determining or, if not, finding good estimates. If we are able to determine , we also consider the problem of characterizing the extremal colorings, that is, in finding .
To determine if a graph is omnitonal is a problem within the scope of interpolation theorems in graph theory; as we shall see, we will incorporate proof techniques typical to this area together with techniques typical in Ramsey theory and extremal graph theory to obtain results concerning omnitonal graphs.
As in the case of balanceable graphs, to prove that a graph is not omnitonal, it is enough to exhibit infinitely many values of for which there is a edge coloring of with the same or almost the same (differing by at most one unit) number of red and blue edges without an colored copy of for some and such that .
Observe that, if a graph is omnitonal, then it is balanceable (by choosing and such that in case (mod ) and in case (mod )) but not necessarily vice versa. For instance, is balanceable (see Theorem 3.1) but the following construction shows that is not omnitonal: Consider a partition of the vertex set and a coloring of the edges such that all edges inside are colored red, all other edges are colored blue and we choose and so that and are equal (this can be done for infinitely many values of , see Lemma 2.3). Evidently, there are no colored copies of in this coloring. Moreover, concerning for arbitrary , observe that the pattern can never be realized for a copy of in the coloring of given above, implying that no complete graph is omnitonal.
The connection to the zerosum problem with weightings is explained below.
Remark 1.3.
Let and be positive integers with and let be an omnitonal graph with (mod ). Then, for large enough , any coloring with contains a zerosum copy of (that is, a copy of where ). To see this, define another coloring where iff and iff . Since is omnitonal, and is such that , we can find an colored copy of with and . Then is a zerosum copy of under coloring :
We are unaware of a systematic study along the lines suggested by the omnitonal graphs, which we start here.
1.3 Results
After this introductory part, the article is divided into four sections. In Section 2, we establish structural results for both balanceable and omnitonal graphs. In Corollary 2.5 and Theorem 2.6, we give necessary and sufficient conditions for a graph to be balanceable and, respectively, omnitonal. The characterization of balanceable graphs is actually a particular case of Theorem 2.4 which provides necessary and sufficient conditions for a graph to be tonal (see Definition 2.2). From these characterizations we derive easily, for example, that trees are omnitonal and, therefore, also balanceable graphs. All these results are consequences of Theorem 2.1, proved in Section 2.1, which is a new version of a result by Cutler and Montágh [10] solving a conjecture proposed by Bollobás (see [10]).
When considering the problem of determining if a graph is balanceable or omnitonal, the study of two particular colorings of , which we will call type (where one color forms a clique of order ) and type (where one color forms two disjoint cliques of order ) colorings, arises naturally. It was shown in [10] that, for sufficiently large , every edgecoloring of with a positive fraction of edges of each color contains a copy of a type or a type colored . Without seeking a sharp bound on , our result prescinds from the quadratic amount of edges of each color, and replaces it with a subquadratic constraint, implying that, in case of existence, and are always subquadratic as functions of . The nature of our proof of Theorem 2.1 avoids probabilistic arguments and relies on the Ramsey Theorem, the Turán numbers and the Zarankiewicz numbers. It also prevents to get good upper bounds on and , which are left for further research to be improved.
Another major idea in this work, presented in Section 2.2, is the study of a class of graphs called amoebas (see Definition 2.11). These graphs were developed here along the proof techniques used in interpolation theorems in graph theory, building upon ideas from [8] and [9]. In particular, we prove that every amoeba is balanceable (Theorem 2.15) and that every bipartite amoeba is omnitonal with and , where and are the Turán number for and the corresponding family of extremal graphs (Theorem 2.14).
In Section 3, we determine as well as for paths and stars with an even edge number. Further, in Section 4, we determine and for stars. Moreover, we show that for every tree on edges. Since paths are bipartite amoebas, and are already covered in Section 2.2.
Finally, in Section 5, we discuss further variants of these concepts and present several open problems.
2 Structural results
2.1 Characterization of balanceable and omnitonal graphs
Let and be integers with . A edgecolored complete graph is said to be of type if the edges of one of the colors induce a complete graph , and it is of type if the edges of one of the colors induce a complete bipartite graph . If , we eliminate the parameter and write for short type and type colorings.
For a given graph , we denote by the color Ramsey number, that is, the minimum integer such that, whenever , any coloring contains either a blue or a red copy of . For a given graph , we denote by the Turán number for , that is, the maximum number of edges in a graph with vertices containing no copy of . The wellknown KővariSósTurán theorem [15] implies that, for the balanced complete bipartite graph ,
(1) 
For a given positive integer , we denote by the Zarankiewicz number, that is, the maximum number of edges in a bipartite graph with vertices in each part, containing no copy of . Here again, the KővariSósTurán theorem yields the following upper bound for :
(2) 
The following Theorem 2.1 is a new version of a result first proved by Cutler and Montágh [10] solving a conjecture raised by Bollobás (see [10]) about the existence, for sufficiently large, of a type or type colored in every edgecoloring of with a positive fraction of edges of each color. The bound on the Ramsey parameter concerning this problem was further improved by Fox and Sudakov in [12]. In both papers, the authors explicitly assume for some and estimate an upper bound on the smallest for which this balancing forces a type or a type colored copy of . Not seeking a sharp bound on , our result, in contrast, prescinds from the balancing restriction and replaces it with a weaker constraint, which in turn allows us to give a subquadratic bound, as a function of , on the minimum number of edges of each color required on the edge coloring of the . The proof of our version avoids probabilistic arguments and uses only the classical Ramsey and Turán numbers for complete bipartite graphs and the Zarankiewicz numbers instead.
Theorem 2.1.
Let be a positive integer. For all sufficiently large , there exists a positive integer and a number such that any coloring with contains a type or a type colored copy of .
Proof.
Let be an integer such that
(3) 
and set ). Now define
which, by (18), is clearly . Assume to be large enough such that we can take a coloring with . This is possible since . By definition, there is a monochromatic, say red, copy of in . Let be a vertex set partition of such a red copy of , where and all edges between and are red. Consider now the complete graph obtained from by removing the vertex set . Since we lose at most 2 blue edges, by the definition of , there are at least blue edges in . Hence, there is a blue copy of in . Let be a vertex set partition of such a blue copy of , where and all edges between and are blue. Observe that the red and the blue copies of that we obtain are vertex disjoint.
Now consider the edge colored graph induced by . By the definition of , we know that there are monochromatic copies of inside both and . If at least one of these monochromatic copies of is blue then, since all edges between and are red, we will have a copy of which is either of type or of type ; and since we are done in this case. Otherwise, we get two red monochromatic copies of , one inside an the other inside , which indeed is a monochromatic, red, copy of . Similarly, by looking at the edge colored graph induced by , either we are done or we get a blue copy of . Hence, we can assume that we have two vertex disjoint monochromatic copies of , one red and one blue. Call the set of vertices of the red one, and the set of vertices of the blue one.
Finally, we consider the edge colored complete bipartite graph, , induced by . Clearly, one of the colors, say red, has at least half of the edges. In other words, there are at least red edges in . By computing the upper bound (2) of the Zarankiewicz number , we obtain the left hand of (3). Thus, by the definition of , we gain a monochromatic copy of in . That is, there are subsets and , with , such that all edges between and are red. Observe that the edge colored complete graph induced by is of type , which completes the proof. ∎
As we said before, for a given graph , Ramsey’s Theorem guaranties, for large enough , the existence of either a colored copy or a colored copy of in every coloring of , while, to force the existence of other color patterns, there also have to be enough edges from each color. The precise amount of edges from each color needed to this aim, if it exists, is the parameter that we are interested in. The following definition extends Definition 1.1 from a balanced proportion of the colors to other proportion variations.
Definition 2.2.
Let be a graph and an integer with . Let be the minimum integer, if it exists, such that every coloring with contains, either an colored copy of , or an colored copy of . If exists for every sufficiently large, we say that is tonal.
Observe that, if exists, then . Clearly this happens, too, for omnitonal graphs: .
Theorem 2.1 allows us to give a characterization of tonal (thus also balanceable) graphs and omnitonal graphs. We will use the following lemma that follows directly from Lemmas 3.1 and 3.2 given in [8].
Lemma 2.3 ([8]).
For infinitely many positive integers , we can choose in a way that the type coloring of is balanced. Also, for infinitely many positive integers , we can choose in a way that the type coloring of is balanced.
Theorem 2.4.
Let be a graph and let be an integer with . Then is tonal if and only if has both a partition and a set of vertices such that .
Proof.
Suppose that is tonal. Let be large enough such that exists and chosen such that there is a balanced type coloring of for some , which is possible by Lemma 2.3. Suppose, without loss of generality, that the graph induced by the red edges in such a coloring of is isomorphic to . Since is tonal and , there must be a copy of in with or red edges, implying that there is a set with . Analogously, we take now an large enough such that exists and chosen such that there is a balanced type coloring of for some , which, again, is possible by Lemma 2.3. Suppose, without loss of generality, that the graph induced by the red edges in such a coloring of is isomorphic to . Since is tonal and , there must be a copy of in with or edges, implying that there is a partition with .
Conversely, suppose that has both a partition and a set of vertices such that . Let be an edge coloring of with , where and is like in Theorem 2.1. Hence, for sufficiently large, there is a type or a type copy of . If this copy is of type , say we have one red and one blue and all edges in between are blue, then we can find a copy of placing the set inside the red and the other vertices inside the blue . If this copy is of type , say we have two red ’s joined by blue edges, then we can find a copy of placing the edge cut such that and are each in one of the red ’s. ∎
Since a graph is balanceable if and only if it is tonal, the following corollary is immediate from Theorem 2.4.
Corollary 2.5.
A graph is balanceable if and only if has both a partition and a set of vertices such that .
Adopting the same proof technieque from Theorem 2.4, we obtain the next result.
Theorem 2.6.
A graph is omnitonal if and only if, for every integer with , has both a partition and a set of vertices such that .
Proof.
Suppose that is omnitonal. Let be large enough such that exists and chosen such that there is a balanced type coloring of for some , which is possible by Lemma 2.3. Suppose, without loss of generality, that the graph induced by the red edges in such a coloring of is isomorphic to . Since is omnitonal and , there must be a copy of in with red edges for every . This implies that there is a set with for every . Analogously, we take now an large enough such that exists and chosen such that there is a balanced type coloring of for some , which, again, is possible by Lemma 2.3. Suppose, without loss of generality, that the graph induced by the red edges in such a coloring of is isomorphic to . Since is omnitonal and , there must be a copy of in with edges for every . It follows that there is a partition with for every .
Conversely, suppose that has both a partition and a set of vertices such that for every . Let be an edge coloring of with , where and is like in Theorem 2.1. Hence, for sufficiently large, there is a type or a type colored copy of . If this copy is of type , then there are two possibilities: either we have one red and one blue and all edges in between are blue or the colors are reversed. In the first case, we can use a set with to find a copy of with red edges and blue edges. In the second, we can use a set with to find a copy of with red edges and blue edges. The case of having a type colored copy of is similar. ∎
Having determined the structure of tonal and omnitonal graphs in Theorems 2.4 and 2.6, we learn from Theorem 2.1 that , thus also , and are all subquadratic as functions of .
Corollary 2.7.
If is an tonal graph, then , where depends only on (this holds true, in particular, for balanceable graphs). Also, if is omnitonal, then , where depends only on .
The next result concerns the chromatic number of omnitonal graphs.
Theorem 2.8.
Omnitonal graphs are bipartite.
Proof.
Take an large enough such that exists and chosen such that there is a balanced type coloring of for some , which is allowed by Lemma 2.3. Suppose, without loss of generality, that the graph , induced by the red edges in such a coloring of , is isomorphic to . Since is omnitonal and , there must be a red copy of contained in the red graph , which is bipartite. Hence, must be bipartite. ∎
Remark 2.9.
There are bipartite graphs which are not balanceable and hence neither omnitonal. For example, cycles of lenght do not appear balanced in any type colored . Moreover, balanceable graphs are not necessarily bipartite: is an example (see Theorem 3.1).
In contrast with the fact that certain even length cycles are not balanceable as given in the above remark, the situation for trees (and for forests) is completely different.
Theorem 2.10.
Every tree is omnitonal.
Proof.
Let be a tree. According to Theorem 2.6, we have to verify that, for every integer with , has both a partition and a set of vertices such that . We proceed by induction on . If then both conditions are clearly satisfied for . Let be a tree with , and let be a leaf where is the only vertex of adjacent to . By the induction hypothesis, the tree satisfies that, for every , there are both a partition and a set of vertices such that . Note that for every the subset satisfies . Likewise, for every we can obtain a partition with by taking and if , or and if . To show that there are both a partition and a set of vertices such that is trivial and the proof is concluded. ∎
It is not difficult to see that the disjoint union of two omnitonal graphs is again an omnitonal graph. Hence, it follows directly from Theorem 2.10 that every forest is omnitonal.
2.2 Amoebas
In this section, we describe a class of graphs which we call amoebas. We are interested in such graphs since, as we shall see below, amoebas are balanceable and provide a wide family of omnitonal graphs, too.
Given a graph of order embedded in a complete graph , where , we say that (also embedded in ) is obtained from by an edgereplacement, if for some and , . Isolated vertices will play no role here, so all graphs considered further on may be the ones induced by its corresponding edge set.
Definition 2.11.
A graph is an amoeba if there exist , such that for all and any two copies and of in , there is a chain such that, for every , and is obtained from by an edgereplacement.
For example, it is not hard to see that a path is an amoeba for every , while a cycle is not an amoeba for any .
The following is a basic interpolation lemma for amoebas.
Lemma 2.12.
Let be an amoeba and consider a coloring where . Let be integers such that and and . If there are both, an  and an colored copy of , then, there is an colored copy of for all integers and such that , and .
Proof.
Under the hypothesis of the lemma, let be an colored copy of , and be an colored copy of with and . Since is an amoeba, and , we know there is a chain such that, for every , and is obtained from by an edgereplacement. Let be the number of red edges in , and be the number of blue edges in , so that, for every , is an colored copy of . Observe that an edgereplacement modifies the color pattern in at most one unit, that is, for every , as well as and . Thus, if we start with an colored copy of , and we end with an colored copy of , we must cover all color patterns with and . ∎
Remark 2.13.
Since, by the KővariSósTurán theorem [15], for any bipartite graph , we have, for large enough , . This means that we can consider colorings with if is sufficiently large.
Note that Lemma 2.12 implies that, for a given amoeba and a given coloring , where , if we can find both a colored copy and an colored copy of , then the so colored will contain the graph in every possible color pattern for and with , and . Therefore, by means of Lemma 2.12 and Remark 2.13, we can prove our next theorem.
Theorem 2.14.
Every bipartite amoeba is omnitonal with and , provided is large enough to fulfill and .
Proof.
Let be a bipartite amoeba. By Remark 2.13 we can consider, for sufficiently large , colorings of with and at least edges of each color. Since, any coloring with contains a colored copy of and an colored copy of , by Lemma 2.12, there is an colored copy of for all integers and such that and . Thus, is omnitonal and . In order to see that , notice that we can give a coloring of with such that there are no colored copies of , and therefore cannot be omnitonal. Further, observe that the fact that implies that . Suppose now there is a graph and let be a coloring of the edges of such that . Then but, since , contains a subgraph isomorphic to , that is, there is an copy of contained in the colored . Since , clearly and there is also a copy of in . Hence, by Lemma 2.12, there is an copy of for every pair of nonnegative integers with , a contradiction to the hypothesis that . Therefore, . ∎
Since the balanceable property is not as restrictive as the omnitonal property, we will see that we can prescind from the bipartite condition to prove that every amoeba is balanceable. For the proof, we will make use of an old argument of Erdős which states that every graph has a bipartition such that (see Lemma 2.14 in [13]). Deleting edges if necessary, one can easily see that every graph contains a bipartite subgraph with .
Theorem 2.15.
Every amoeba is balanceable.^{2}^{2}2Observe that the proof of this theorem yields actually that every amoeba is strongly balanceable in the sence discussed in the footnote of Definition 1.1.
Proof.
Let be an amoeba. By the observation above, we may consider a bipartite subgraph of having exactly edges. Let be a coloring with , which is possible for large enough because of Remark 2.13. Hence, we know that contains a colored copy of and a colored copy of . Now we can complete those copies of into copies of in an arbitrary way to get an colored copy of , and an colored copy of , where and . Since , we also have and . Altogether we have and . Hence, Lemma 2.12 implies that contains a copy and a copy of . ∎
Remark 2.16.
Observe that not every amoeba is bipartite. For example, one can easily check that odd cyles together with a pendant vertex are nonbipartite amoebas. Moreover, there are also omnitonal graphs which are not amoebas. For instance, due to the fact that every tree is omnitonal (Theorem 2.10), stars with leaves are omnitonal, too, but it is evident that they are not amoebas.
Amoebas are interesting not only because of their good behavior concerning balanceable and omnitonal graphs, but we think they are interesting for their own. A forthcoming paper under preparation [7] will deal with such an analysis.
3 Balanceable graphs
The study of balanceable graphs (in disguise) has already been started in the following three recent papers. The first one is a paper by Caro and Yuster [6], where zerosum weighting over are introduced and several zerosum theorems are proved that fit to the framework of balanceable graphs as explained above. The other two [8, 9] develop further the study on weightings on the set of positive integers or on the set of edges of , forcing zerosum copies of given structures (blocks of consecutive integers in the first case, copies of graphs in the second case) which can be translated into the language of colorings and balanceable graphs.
We restate here, in the language of redblue coloring, instead of weighting and zerosum language, a part of the main theorem from [9], which is a sort of rolemodel for the results in this section.
Theorem 3.1 ([9]).

For any positive integer , ,