Given a graph , let denote the number of vertices and denote the number of edges.111In this paper, all graphs we consider are finite, undirected and simple, unless otherwise specified. A cut of is a bipartition of the vertices , and the size of a cut is the number of edges between and . The of a graph , denoted , is the size of largest cut of .
There has been extensive work understanding the from an extremal perspective. Most simply, by taking a random cut, we see that every graph with edges has value at least . Edwards [11, 12] showed that every graph with edges satisfies for some , which is tight up to a choice of . In this article, we adopt a similar perspective but study families of graphs equipped with additional structure, namely having a fixed forbidden subgraph , and parametrize our bounds by a measure of sparseness called degeneracy.
We say a graph is -free if it does not contain as a subgraph. Let be the minimum of an -free graph on edges. The quantity has been studied extensively. Alon [1, 2] studied the of triangle-free graphs, showing that . Alon, Krivelevich and Sudakov  showed tight bounds on for a number of sparse graphs . When is a tree on vertices, they showed that where is 2 if is even and 0 if is odd; they further observed that equality holds for infinitely many . For even cycles , they showed that and that this bound is tight up to a choice of constant in the lower order term for 4-cycles, 6-cycles, and 10-cycles. They also considered complete bipartite graphs, showing that and that . They also showed that, for obtained by connecting a vertex to a nontrivial forest, . Recently, Zeng and Hou  considered complete graphs, showing that for all .
Previous work has also considered the complement of the problem, i.e. the minimum number of edges that must be removed to make a graph bipartite, or more generally -partite. A longstanding conjecture of Erdős (he wrote in 1975  that it was already old) states that every triangle -free graph on vertices can be made bipartite by deleting at most edges. If true, this conjectured bound is the best possible: this can be seen by considering a balanced blow-up of a cycle on five vertices. While this problem has been seriously investigated, the best known upper bound  is approximately , and thus Erdős’s conjecture remains open. Solving a different conjecture of Erdős, Sudakov  showed that any -free graph on vertices can be made bipartite by removing at most edges. This bound is tight, which can be seen by considering a balanced blow-up of a triangle. Sudakov further conjectured for that the balanced complete -partite graph on vertices is the furthest from being bipartite over all -free graphs. A recent result  showed that any -free graph on vertices is at most edges from bipartite.
The results in the previous two paragraphs are more useful for graphs with many edges, and give much weaker bounds for sparse graphs. One might hope to give bounds on the in terms of some sparseness property of the graph, like the maximum degree or degeneracy, and in this article, we address this question for a variety of choices of .
A -degenerate graph is a graph such that every induced subgraph has a vertex of degree at most . Equivalently, is -degenerate if there exists an ordering of the vertices such that every vertex has at most neighbors with index . Degeneracy is a broader notion of sparseness than maximum degree: all maximum degree graphs are -degenerate, but the star graph is 1-degenerate while having maximum degree .
Let be the minimum of a -degenerate -free graph with edges. We largely focus on the case is a clique on vertices, but also give bounds on for several other families of forbidden subgraphs , including odd wheels (obtained by connecting a central vertex to each vertex of an even cycle ), the complete bipartite graphs , and cycles .
Some bounds are known for the of -degenerate graphs. The expected of a random -regular graph
is, with high probability,. For arbitrary -degenerate graphs ,
a bound met up to the constant by the disjoint union of ’s. This bound can be obtained by randomly ordering the vertices of and greedily adding them to a constructed cut one at a time from that ordering. The expected number of vertices with an odd number of neighbors before it is , which is at least (as is -degenerate). Each such vertex increases the difference between the number of cut and uncut edges by at least , giving a cut of size . Shearer  gave a tight (up to a constant factor) bound on for -free -degenerate graphs, showing that there exists such that, for all .
This bound is tight up to choice of , seen by taking a random -regular graph and removing an edge from every triangle (for details, see e.g. Proposition 1.3). In the case that graph has maximum degree , Shearer’s bound was generalized by Carlson, Kolla, and Trevisan to -free graphs, who showed the following result.
Theorem 1 ().
There exists a such that for all and all , every -free graph on edges and maximum degree satisfies
We improve on the above bound in two ways. First, we generalize from maximum degree to -degenerate, and second, we improve the exponent of in the lower order term.
There exists a such that for all and , we have
For maximum degree graphs, Theorem 2 matches Theorem 1 when and gives a strict improvement over Theorem 1 when (up to the constant ). In the special case , we modify our method to improve the exponent in from to .
There exists a constant such that for all , we have
To prove Theorem 2 and Theorem 3, we make use of a more general framework for lower bounds described in Section 2.3. These methods also allow us to leverage the bounds in  to give nontrivial lower bounds on the of -degenerate -free graphs for several families of sparse forbidden subgraphs .
For a graph , let
When is one of the above, there exists such that, for all
Note that the case forest+1 includes the cases when is a cycle and when . While Theorems 2 and 3 improve on Theorem 1, we show, using the same methods, that a stronger lower bound is true assuming the following conjecture of Alon, Bollobás, Krivelevich, and Sudakov .
Conjecture 1.1 ().
For any graph , there exists constants and such that for all ,
Assuming Conjecture 1.1, we show that the exponent of in the lower order term of is bounded away from for every graph . Qualitatively, this contrasts with Theorems 1 and 2, where the exponents approach as increases.
Assuming Conjecture 1.1, for any graph , there exist constants and such that, for all , we have
With these results in mind, we pose the following conjecture.
For any graph , there exists a constant such that, for all , we have
Theorem 4 shows that Conjecture 1.2 is true when is a forest with a common neighbor (up to logarithmic factors in the lower order term). To disprove Conjecture 1.2, one would need to construct -degenerate graphs on vertices with at most . However, Turan’s theorem implies that a -free graph with edges has at least vertices. For , the Erdős -Rényi graph with high probability (as ) satisfies
is the smallest eigenvalue of the graph. In-regular graphs with , by Alon-Boppana  theorem, this bound cannot be smaller than .
Alon, Krivelevich, and Sudakov  showed that, when is a forest, the of -free graphs is for some independent of . This result holds independently of the density of the graph, and in particular also applies to -degenerate graphs, where the constant in the lower order term is independent of . For -degenerate graphs, we observe that forests are the only graphs for which this is true: whenever contains a cycle, there exist infinitely many -free -degenerate (and, in fact, maximum degree ) graphs on vertices with no larger than . In particular, Conjecture 1.2 is optimal (up to a constant depending on in the lower order term) if it is true when is not a forest.
For all and , there exist and such that for all , there exists a -free graph on vertices with maximum degree and
In Table 1, we summarize our lower bounds on for -free graphs and how they compare to those in the literature.
|Forbidden subgraph||Prior work||This work||Tight?|
|Thm 5 if Conj 1.1|
|for odd||Thm 4|
Concurrent work by Sudakov
Organization of paper
In Section 2, we present a general framework to convert lower bounds on the in general (denser) graphs to the of -degenerate graphs. In particular, we show how to convert bounds on to bounds on . In Section 3, we apply the results in Section 2 to obtain improved bounds on for , proving Theorems 2, 3, and 5. In Section 4, we apply the results in Section 2 to obtain bounds on for a variety of forbidden subgraphs , proving Theorem 4. In Section 5, we prove Theorem 6, showing that Conjecture 1.2 implies Conjecture 1.1. In Section 6, we construct cycle-free graphs from random -regular graphs with small , proving Proposition 1.3. We conclude with some remarks and further directions in Section 7.
2. in -degenerate graphs
To prove Theorems 2, 3, 4, and 5, we adapt methods historically used to give bounds in general graphs to give meaningful lower bounds on the of -degenerate graphs. In other words, we are converting bounds on to bounds on (except in Theorem 2, where we do something slightly better).
To generalize from the setting of degree bounded graphs, we make use of some helpful notation. Give a graph and a subset of vertices , we let denote the subgraph induced by vertices , and we let be shorthand for , the number of edges in . We also let denote the number of triangles of .
Given a graph with , an ordering is a bijection . With respect to some ordering , let be the number of neighbors of such that .
By definition, every -degenerate graph has an ordering for which for all .
2.1. in Triangle-Deficient Graphs
We first show a lower bound that arises from the SDP relaxation of , formulated below for a graph :
The Goemans-Williamson  rounding algorithm is a classical rounding algorithm for
that gives an integral solution from a vector solution. This rounding was used in to lower bound the of a maximum degree graph with few triangles, and we extend their approach to -degenerate graphs.
Let . Let be a -degenerate graph with edges and triangles. Then
Since is -degenerate, there exists an ordering of the vertices such that for all , we have . For define by
For , let . By the definition of the -degenerate ordering, we have for all . For edges with , we have
For , we observe that is at most if vertices form a triangle in and otherwise. For , we have as . Thus, for all edges with ,
where denotes the number of indices with such that form a triangle.
Vectors form a vector solution to the SDP (2.1). We now round this solution using the Goemans-Williamson  rounding algorithm. Let denote a uniformly random unit vector, , and . Note that the angle between vectors is equal to , so the probability an edge is cut is
In the last inequality, we used that, for , we have . This is true as when is positive and when is negative. Thus, the expected size of the cut given by is, by linearity of expectation,
The equality holds because counts each triangle of exactly once. ∎
Lemma 2.2 gives the following immediate corollary.
There exists an absolute constant such that the following holds. For all and , if a -degenerate graph has edges and at most triangles then
2.2. Decomposing degenerate graphs
Graphs that are -free have fewer than the expected number of triangles of a random graph of similar density. Carlson, Kolla, and Trevisan (Claim 4.3 of ) noted that maximum-degree graphs with few triangles must have small subsets of neighborhoods with many edges. We give a -degenerate generalization of this lemma.
Let and , and let be a -degenerate graph with at least triangles. Then there exists a subset of at most vertices with a common neighbor in such that the induced subgraph has at least edges.
Since is -degenerate, we fix an ordering of the vertices such that for all . Then, if denotes the number of triangles of where , we have
Hence, there must exist some such that . Let denote the neighbors of with index less than . By definition, the vertices of have common neighbor . Additionally, has at least edges and vertices, proving the lemma. ∎
We can use this bound to describe as the union of a collection of subgraphs with helpful properties. The following lemma was proven implicitly in  for graphs with maximum degree , and we generalize it to -degenerate graphs.
Let . Let be a -degenerate graph on vertices with edges. Then there exists a partition of the vertex set with the following properties.
For , the vertex subset has at most vertices and has a common neighbor, and the induced subgraph has at least edges.
The induced subgraph has at most triangles.
We construct the partition iteratively. Let . For , we partition the vertex subset into as follows. If has at least triangles, then by applying Lemma 2.4 to the induced subgraph , there exists a vertex subset with a common neighbor in such that and the induced subgraph has at most edges. In this case, let . Let denote the maximum index such that is defined, and let . By construction, satisfy the desired conditions. By definition of , the induced subgraph has at most triangles, so for , we obtain the desired result. ∎
2.3. Large from smaller forbidden subgraphs
In , the authors obtain a partition of similar to that Lemma 2.5, such that the induced subgraphs are all -free. They recursively bound the of these smaller -free induced subgraphs , applying a version of Corollary 2.3 to bound the of . Finally they combine the cuts randomly to obtain a cut of .
While we follow a similar approach at the outset, we observe that in the partition, graphs all have at most vertices. Thus we can obtain a stronger bound on the of these induced subgraphs by applying known results about the of more general, dense graphs.
Towards our goal of obtaining tighter bounds on , we show how to leverage existing bounds on the in general graphs to obtain bounds in the -degenerate setting by finding subgraphs of that are either small and dense or triangle-deficient, and combining maximal cuts of these subgraphs.
There exists an absolute constant such that the following holds. Let be a graph and be obtained by deleting any vertex of . Let . For any -free -degenerate graph , one of the following holds:
There exist graphs such that five conditions hold: (i) graphs are -free for all , (ii) for all , (iii) for all , (iv) , and (v)
Let be the parameter given by Corollary 2.3. Let .
Let be a -degenerate -free graph. Applying Lemma 2.5 with parameter , we can find a partition of the vertex set with the following properties.
For , the vertex subset has at most vertices and has a common neighbor, and the induced subgraph at least edges.
The subgraph has at most triangles.
For , let and let . For , since is -free and each is a subset of some vertex neighborhood in , the graphs are -free. For , fix a maximal cut of with associated vertex partition . By the second property above, the graph has at most triangles. Applying Corollary 2.3 with parameter , we can find a cut of of size at least with associated vertex partition .
We now construct a cut of by randomly combining the cuts obtained above for each as in . Independently, for each , we add either or to vertex set , each with probability . Setting , gives a cut of . As partition , each of the edges that is not in one of the induced graphs has exactly one endpoint in each of with probability . This allows us to compute the expected size of the cut (a lower bound on as there is some instantiation of this random process that achieves this expected size).
We bound (2.3) based on the distribution of edges in in cases:
. Then, (2.10) holds, as
The number of edges between and is at least . Then, the cut given by vertex partition with and has at least edges, in which case , so (2.11) holds.
Else, must have at least edges. Note that for all , the graph is free, has at most vertices, and at least edges by construction. Since is -degenerate, is as well, so
Hence . Lastly, by (2.3), we have
Lemma 2.6 allows us to convert bounds on to bounds on .
Let be a graph and be obtained by deleting any vertex of . Suppose that there exists constants and such that, for all positive integers , we have . Then there exists a constant such that for all ,
Let be the parameter in Lemma 2.6. We may assume without loss of generality that . Let be a -degenerate -free graph and . Let .
For all , we have
where follows since , follows since and , and follows since . Hence, as , we have
as desired. ∎
3. of -free sparse graphs
We specialize Lemmas 2.6 and 2.8 to the case that to obtain both a lower bound and conditional lower bound on the of a -free graph. Let denote the chromatic number of a graph , the minimum number of colors needed to properly color the vertices of the graph so that no two adjacent vertices receive the same color.
3.1. -free graphs
We obtain a nontrivial upper bound on the chromatic number of a -free graph , giving an lower bound (Lemma 3.4) on the of -free graphs. This lower bound was implicit in , but we provide a proof for completeness. The lower bound on the of general -free graphs enables us to apply Lemma 2.6 to give a lower bound on the of -degenerate -free graphs per Theorem 2. The following well known lemma gives a lower bound on the using the chromatic number.
Lemma 3.1 (see e.g. Lemma 2.1 of ).
Given a graph with edges and chromatic number , we have .
Since , we can decompose into independent subsets . Partition the subsets randomly into two parts containing and subsets , respectively, to obtain a cut. The probability any edge is cut is , so the result follows from linearity of expectation. ∎
Let and be a -free graph on vertices. Then,
We proceed by induction on . For , the statement is trivial as the chromatic number is always at most the number of vertices. Now assume has vertices and that for all -free graphs on vertices. The off-diagonal Ramsey number satisfies