Large subgraphs without short cycles
We study two extremal problems about subgraphs excluding a family of graphs. i) Among all graphs with edges, what is the smallest size of a largest –free subgraph? ii) Among all graphs with minimum degree and maximum degree , what is the smallest minimum degree of a spanning –free subgraph with largest minimum degree? These questions are easy to answer for families not containing any bipartite graph. We study the case where is composed of all even cycles of length at most , . In this case, we give bounds on and that are essentially asymptotically tight up to a logarithmic factor. In particular for every graph , we show the existence of subgraphs with arbitrarily high girth, and with either many edges or large minimum degree. These subgraphs are created using probabilistic embeddings of a graph into extremal graphs.
Let be a simple undirected graph with vertices. If is a given graph, then we say that is –free if there is no subgraph of isomorphic to . The problem of determining the Turán number with respect to and , i.e. the largest size of an –free graph on vertices, has been extensively studied in the literature. This is the same as determining the size of a largest –free spanning subgraph of , the complete graph on vertices. The latter quantity is denoted by . We can extend this notion in a natural way: for every graph , let denote the largest size of an –free subgraph of . Also, for a family of graphs, we say that is –free if does not contain any graph from , and denote by the largest size of an –free subgraph of .
In this paper, we provide lower bounds for in terms of different graph parameters of . If does not contain any bipartite graph, it is easy to provide tight bounds for . Therefore, it is more interesting to study the behavior of when contains bipartite graphs. We mostly address the case of even cycles , with .
In the first part of the paper, we derive a lower bound for in terms of the size of . In the second part of the paper, we study the largest minimum degree of a –free spanning subgraph of in terms of the maximum and minimum degrees of . In both cases, for every graph , we show the existence of subgraphs with either many edges or large minimum degree, and arbitrarily high girth.
As far as we know, this is the first study of these extremal problems.
Key definitions. Let be a fixed graph. We define as the smallest possible size of a largest –free subgraph of a graph with edges, that is,
More generally, for a family of graphs, we define
Our second key definition is , the smallest possible minimum degree of an –free spanning subgraph of a graph with minimum degree and maximum degree such that is maximized. If we define , then we can write
As an illustration, consider the case , the path on three vertices. Since every graph with minimum degree at least 2 contains a copy of , for any and with , we have .
Also, for a family if we define , then
Thomassen’s conjecture. A related problem for the girth and the average degree was stated by Thomassen . Similarly to our definitions of and , we can define the smallest possible average degree of an –free subgraph of a graph with average degree such that is maximized. Formally, if we define , then:
A reformulation of Thomassen’s conjecture is to say that for every fixed , when . If , it is easy to check that . For , Kühn and Osthus  showed that , thus confirming the conjecture for this case.
Observe that one can change the average degree for the minimum degree in the definition of , since a graph with average degree has a subgraph with minimum degree at least . It is interesting to notice that Thomassen’s conjecture is true when restricted to graphs of large minimum degree with respect to the maximum degree . In this case, a relatively straightforward application of the local lemma shows that for any family of graphs there exists a spanning -free subgraph with minimum degree , where .444See Lemma 4 of  for a similar statement with . One of the goals of this paper is to improve this lower bound when is composed of cycles.
Families of non-bipartite graphs. It is rather easy to determine both functions and asymptotically when (there are no bipartite graphs in ), as shown by the following proposition (a proof is provided in Section 2.1).
For every graph and for every there exists a –partite subgraph of such that for every ,
Moreover, for every with , we have
Even cycles. It remains to study and when contains a bipartite graph. In this paper, we focus on the cases and for some . We define
Using Proposition 1 one can easily show that every family composed of and other graphs with chromatic number at least satisfies
This provides us a way to transfer the results obtained for to .
Denoting by the complete bipartite graph with parts of size and , our first result is:
For every there exists such that for every large enough ,
Observe that for every dividing , we have
The case of the above problem appears to be more accessible. In particular, combining the upper bound of Kovári, Sós and Turán  and Reiman , and the lower bound provided by Erdős and Rényi  we have (see Chapter 6.2 of  for a discussion). Here we derive the following corollary (proved in Section 3):
There exists a constant such that for every large enough ,
By using (1) with , Theorem 2 provides a lower bound on , which implies the existence of large subgraphs with high girth. We also provide a general lower bound for , proved in Section 4 (where denotes the girth of graph ):
Let be a graph with minimum degree and (large enough) maximum degree such that for some large constant . Then, for every there exists a spanning subgraph of with and , for some small constant . In particular, under the above conditions on and ,
The existence of a graph with a given number of vertices, many edges and large girth is one of the most interesting open problems in extremal graph theory (see Section in ). The best known result that holds for any value of was given by Lazebnik, Ustimenko and Woldar in . They showed that ). Using that, we can obtain the following explicit corollary of Theorem 4.
For every and such that for some large constant , we have
The upper bound for the Turán number of even cycles , shown by Bondy and Simonovits , implies that
Since the above cited upper bound for is conjectured to be of the right order, Corollary 5 is probably not tight.
For the particular case we can derive a better bound. Erdős, Rényi and Sós  and Brown  showed that for every prime there exists a –free -regular graph with vertices. By the density of primes, for every there exists a graph of order at most satisfying the former properties. Thus, we can obtain the following corollary of Theorem 4:
For every and such that for some large constant , we have
This corollary is tight up to a logarithmic factor, as will be shown in
Proposition 14 (Section 4). The
condition on and is also tight up to a logarithmic factor since, if , any spanning –free subgraph of satisfies .
Similar results can be derived for , since there exist graphs with girth at least and respectively and large minimum degree [18, 16].
For every vertex let denote the set of neighbors of in and . If the graph is clear from the context we will denote the above quantities by and , respectively. We use to denote the natural logarithm of .
2.1 Proof of Proposition 1
Let be a graph with edges and consider a –partition of that maximizes the number of edges in the subgraph . Then, we claim that for every , .
For the sake of contradiction, suppose that there is a vertex , , with degree in less than . Then, there are more than neighbors of in and there is a part , , with at most neighbors of . Moving from to increases the number of edges in by at least one, which gives a contradiction by the choice of .
Clearly, does not contain any copy of with , and thus it does not contain any graph in . Observe also that has at least edges and minimum degree .
Notice that and for every . Let be such that . The upper bound for follows from (3) by choosing for which , and then by taking to be any subgraph of with exactly edges. For the upper bound on (assuming ) take disjoint copies of , add a new vertex and connect it to one vertex from each of the cliques . If a subgraph of the so obtained graph is -free, then the subgraph of spanned by the vertex set of each of the cliques is -free as well, thus implying by (3) that . ∎
2.2 Useful definitions
The following definitions will be useful in our proofs.
For every graph , every with and every vertex labeling we define the spanning subgraph as the subgraph with vertex set where an edge is present if and only if .
For every graph , every with and every vertex labeling we define the spanning subgraph as the subgraph with vertex set such that an edge is present in if all the following properties are satisfied:
, that is ,
for every , , we have , and
for every , , we have .
These two definitions are crucial for our proofs. We will use them to randomly embed a (large) graph into a fixed extremal graph satisfying certain conditions, therefore creating a subgraph of . The way in which we construct the subgraph will allow us to prove that it preserves some of the properties of .
The concept of frugal coloring was introduced by Hind, Molloy and Reed in . We say that a proper coloring is -frugal if for every vertex and every color ,
that is, there are at most vertices of the same color in the neighborhood of each vertex. For instance, a –frugal coloring of is equivalent to a proper coloring of .
2.3 Probabilistic tools
Here we state some (standard) lemmas we will use in the proofs.
Lemma 9 (Chernoff inequality for binomial distributions ).
Let be a Binomial random variable. Then for all ,
Let be a functional. We say that satisfies the Lipschitz condition if for every and differing in just one coordinate from the product space , we have
Lemma 10 (Azuma inequality, Theorem in ).
Let satisfy the Lipschitz condition relative to a gradation of length (i.e. ). Then for all
Lemma 11 (Weighted Lovász Local Lemma ).
Let be a set of events and let be a dependency graph for .
If there exist weights and a real such that for each :
3 Subgraphs with large girth and many edges
Let be a graph with edges. Define to the set of vertices of with degree at least , and . Observe that
and thus spans at most edges. Recall that is the family of all even cycles of length at most . In order to find an –free subgraph with many edges, we will remove all the edges inside and only look at the edges between and , and the edges inside . We will split the proof into two cases in terms of the number of edges between and .
Case 1: . Consider the following partition of into sets , where contains all vertices of whose degree into is between and .
Notice that . Hence, there is a subset incident to at least edges leading to . Let , then and since all the degrees in are between and . Let be the vertices of .
Fix an –free bipartite graph with parts and , and edges, and let be the -coloring defined by for any and by for any , where is chosen independently and uniformly at random.
We say that is good for if:
exactly one neighbor of in is colored by .
Let be the subgraph of with vertex set that only contains the edges , where , , and is good for . We claim that is an –free graph. Assume that there is a cycle of length of even length in with and let be its vertices. Observe that there should be at least one repeated color in the vertices of the cycle, otherwise would contain a . Let () such that and for every , . Besides, since is -frugal in (in the sense that for each vertex of , all its neighbors are colored differently by ), . Thus, there exists a cycle of length at least and at most in , a contradiction since is bipartite and, since it is –free, it has no even cycles of length at most .
Let us compute the expected size of . Fix . Then, the probability that a given edge exists in is
Thus, the expected degree of in is of order
Recall that , hence the expected number of edges of is of order. Since for some small constant , the expected number of edges is of order at least
Case 2: . Then , and all the degrees in are less than . For the sake of convenience we will assume that is an integer.
We will find a large –free graph inside . For this, let be an –free graph on vertices with the largest possible number of edges. Assume and let be a random labeling of the vertices of . Observe that using Proposition 1 we can select a bipartite subgraph of with at least edges (in particular, is –free).
Consider the graph from Definition 8 applied to the induced subgraph . Since is –free, is also –free. For each edge of spanned by , its probability to belong to is at least
since . To justify the above estimate, first choose a label for , then require a label of to be one of the neighbors of in , and finally for each of the neighbors of and in choose a label different from and .
Thus, we expect
edges in the –free subgraph of . ∎
We now prove Corollary 3.
By Theorem 2 we have
for some small constant . By the symmetry of and , we may assume without loss of generality that the minimum is attained when .
We provide two constructions. First, the star of size , which is a –free graph, shows that .
On the other hand, one can construct a –free graph by selecting a subgraph of a larger –free graph. Let be a largest –free subgraph of . We construct a –free bipartite graph by keeping the vertices with highest degrees in one of the parts of . Then is a subgraph of and has at least edges.
Hence, for every ,
4 Subgraphs with large girth and large minimum degree
We devote this section to the proof of Theorem 4. Before proving the theorem, let us state an important observation and an auxiliary lemma.
In this proof we will use a graph that satisfies the following: it has (where ) vertices, girth at least and minimum degree at least .
The existence of such is provided by the following observation.
Let be an -free graph on vertices with edges. Then we claim that has a subgraph satisfying the desired properties. Assume it is not the case, namely all its subgraphs of size at least have minimum degree smaller than . It is also clear that . By iteratively removing vertices of degree smaller than , we can obtain an -free graph on vertices with at least edges, a contradiction.
In a vertex-colored graph, a cycle is called rainbow if all its vertices have distinct colors. A path is called maximal inner-rainbow if its endpoints have the same color , but all other vertices of the path are colored with distinct colors (other than ). We will use the following lemma:
Let be a graph with maximum degree and minimum degree , that admits a –frugal coloring without rainbow cycles of length at most and maximal inner-rainbow paths of length for every . If and is large enough, then there exists a subgraph such that
, , and
Let the color classes of be , , . Assign to each edge a random variable uniformly distributed in . We construct the following subgraph : for every pair of color classes of , an edge between and is retained in if is less than for every between and incident with . Observe that, by construction, is a -frugal coloring of , that is, the vertices of any pair of color classes of induce a matching.
We claim that deterministically (i.e., with probability 1) . For the sake of contradiction, suppose that and let be a cycle in with . Since does not induce any rainbow cycle of length at most in , there exist at least two vertices of with the same color. Let be such that, , and for every , . Then since is a –frugal coloring in and is not rainbow. Then is a maximal inner-rainbow path with a forbidden length, a contradiction.
Now, it remains to show that with positive probability the obtained subgraph has the desired minimum degree.
Observe first that a given edge , with and , is preserved in with probability
For every consider the random variable equal to the degree of in . We have: . By applying Azuma’s inequality (Lemma 10) we will now show that with probability exponentially close to 1, is large enough.
First of all, observe that only depends on the edges that connect a neighbor of to a vertex of color . Let be the set of these edges:
Since is -frugal, we have . Let . Then depends only on a vector in .
If two functions differ only on one edge of , . Thus satisfies the –Lipschitz condition. Let be the event that . Since the martingale length is at most , by setting in Lemma 10,
Since is influenced only by the edges in , it is thus independent of each unless is at distance at most 4 from , and there are at most such events.
since . Thus, by the Lovász Local Lemma (Lemma 11) with and we have
and thus there is a way to assign values to so that . ∎
Now we are ready to prove Theorem 4.
The idea in this proof is to randomly color the vertices from with colors, where for the graph introduced above. We then consider the subgraph from Definition 7 induced by the coloring and the graph . We will show that with positive probability, such a coloring is -frugal in and that contains neither rainbow cycles of length at most nor maximal inner-rainbow paths of length . The value of will be set later in the proof. Then, we will use Lemma 13 to obtain the desired subgraph.
Let be a uniformly random coloring of with colors. Consider the -free graph of order that satisfies and provided by Observation 12. Then construct the spanning subgraph of .
Since , does not induce any rainbow cycle of length at most in , nor any maximal inner-rainbow path of length at least 3 and at most . Moreover, is a proper coloring of , since has no loops.
We will use the Lovász Local Lemma to show that there is a positive probability that the random -coloring of satisfies the following properties:
for every , and
is -frugal in .
For this purpose we will define the following events:
Type A: for each , is the event ,
Type B: for each vertex and each set , is the event , for every distinct .
Observe that the probability that an edge is retained in is at least . Thus, . Using Part 1 of Lemma 9 with , one can check that