Chromatic number, induced cycles, and non-separating cycles

Chromatic number, induced cycles, and non-separating cycles Hanbaek Lyu1

arXiv:1211.0627v6 [math.CO] 5 Oct 2016

Abstract We show that connected graphs with large chromatic number must contain a large number of induced cycles, and that 3-connected graphs with large clique minor contains many induced non-separating cycles, which are known to generate the cycle space of any 3-connected graph G = (V, E ) by a classic theorem of Tutte. This theorem in particular gives a lower bound of |E | − |V | + 1 on the number of such cycles. As an application of our result, we give a new lower bound on the number of induced non-separating cycles in 3-connected graphs of order O(k 3 log3/2 k + k|V |) where k = |E |/|V |. We also give a short proof to the above mentioned theorem of Tutte. Keywords: Chromatic number, induced cycles, Hadwiger number, induced non-separating cycles, 3-connected, Euler characteristic

1. Introduction In this paper, every graph is finite and simple. If S, H are vertex-disjoint subgraphs of G, then S + H is defined by the subgraph obtained from S ∪ H by adding all the edges in G between S and H . On the other hand, S − H := S − V (H ) denotes the subgraph of G obtained from S by deleting all vertices of H and edges incident to them. If v ∈ V (H ) and H is a subgraph of G, then N H (v) denotes the set of all neighbors of v in H and degH (v) := |N H (v)|. A graph G is 3-connected if |V (G)| ≥ 4 and G − H is connected for all subgraph H such that |V (H )| ≤ 2. A subgraph C of a graph G is called a cycle if it is connected and 2-regular, meaning degC (v) = 2 for all v ∈ V (C ). A cycle C ⊂ G is induced if H ⊂ C for any subgraph H ⊂ G such that V (H ) = V (C ), and non-separating (or peripheral) if G −C is connected. C (G) (resp. F (G)) denotes the set of all induced (resp. induced non-separting) cycles in G. A graph parameter is a complexvalued function defined on the isomorphism classes of graphs. Given a graph G = (V, E ), an embedding of G into an orientable surface Πg is a drawing of G in Πg without distinct edges crossing each other at an interior point. The genus Γ(G) of G is the least integer g such that G can be embedded in Πg . We say G is planar if it can be embedded in a plane, i.e., Πg = 0. Given a graph G embedded into a surface Πg , the simply connected regions on the surface bounded by the vertices and edges are called faces. A classic theorem of Euler says that |F(G)| − |E (G)| + |V (G)| = 2 − 2g (1) where F(G) is the set of all faces. The graph parameter on the left hand side of the above equation is called the Euler characteristic. Graph parameters obtained by replacing the set F(G) of faces in the Euler charactersitic by a certain set C(G) of cycles called cycle double cover were ∗ Department of Mathematics, The Ohio State University, Columbus, OH 43210, [email protected]

consider by Vince and Littel [7] and Yu [8] in order to study discrete Jordan curves. We take C(G) to be C (G) or F (G) and study monotonicity properties of the corresponding graph parameters under edge or non-edge contractions. This gives us new inequalities relating various quantities such as the chromatic number and Hadwiger number with the number of induced cycles or induced non-separating cycles. First, we consider the graph parameter Λ(G) under edge contractions. Namely, for a given graph G = (V, E ) and an edge uv ∈ E let G/uv be the graph obtained by identifying vertices u and v into a single vertex v e . This operation G → G/uv is called contraction, or sometimes strong contraction to distinguish it from topological contraction; here when we contract the edge uv, we automatically suppress any multiple edges or loops that might occur after the ordinary topological contraction so that the resulting graphs are always simple. We say a graph H is a minor of G if it can be obtained by a sequence of contractions from a subgraph G 0 ⊂ G. We write H ≤m G if H is a minor of G. The Hadwiger number of G, denoted h(G), is the maximum number n such that the complete graph K n with n vertices is a minor of G. Theorem 1. Let G = (V, E ) be a 3-connected graph and let e ∈ E be such that G/e is 3-connected. Then for a graph parameter Λ(G) = |F (G)| − |E (G)| + |V (G)|, we have Λ(G/e) ≤ Λ(G)

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By using Seymour’s Splitter theorem [4], one can find a sequence of edge contractions from G to its maximal clique minor K h(G) such that every intermediate graph is 3-connected. So we have the following corollary Corollary 2. Let G = (V, E ) be a 3-connected graph. Then we have à ! à ! à ! h(G) h(G) h(G) − + ≤ |F (G)| − |E (G)| + |V (G)| 3 2 1

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Tutte [6] showed that the induced non-separating cycles in a 3-connected planar graph drawn in a plane are exactly the boundaries of faces. Hence Λ(G) = 2 when G is 3-connected planar, so that Corollary 2 implies h(G) ≤ 4. Even though this is a trivial statement, this observation tells us that the exact counterpart of Corollary 2 for the chromatic number implies the famous 4-color theorem, that χ(G) ≤ 4 for (3-connected) planar G. As the Hadwiger number is defined upon the graph operation of contraction and taking minors, the chromatic number is defined in terms of homomorphisms, which can be thought of “dual” operations to the previous ones. This motivates us to study a similar graph parameter and establish its monotonicity properties under homomorphisms. Unfortunately, in contrast to the extensive literature about 3-connectedness and induced non-separating cycles in relation to contractions and minors, we lack a matching level of understanding of them in relation to homomorphisms. Instead, we prove a (much) weaker result; we enlarge F (G) to the set C (G) of all induced cycles in G and prove similar results. Given two graphs G and H , a map φ : V (G) → V (H ) is called a homomorphism if it preserves the adjacency relation, i.e., uv ∈ E (G) implies φ(u)φ(v) ∈ E (H ). We adapt the convention of writing φ : G → H for homomorphism φ : V (G) → V (H ). For example, identifying two nonadjacent vertices is a homomorphism, sometimes called a nonedge contraction. Another example of homomorphism is the vertex coloring; a n-coloring of G is a homomorphism G → K n where K n is the clique of n vertices. The chromatic number of G, denoted by χ(G), is the least integer n such that G admits a n-coloring. One can think of the coloring G → K χ(G) as a sequence of nonedge contractions from G to a smallest complete graph. Now we introduce a notion of locally optimal coloring. Given a graph G and a vertex w ∈ V , let G/w be 2

the graph obtained by identifying pairs of nonadjacent vertices in N (w) such that the homomorphism restricted on the induced subgraph Γ := G[N (w)] is a χ(Γ)-coloring. We call the homomorphism G → G/w a neighborhood compression. Note that G/w may not be uniquely defined. Theorem 3. Let G = (V, E ) be a connected graph and G → G w a neighborhood compression for any vertex w ∈ V . Then for a graph parameter Λ(H ) = |C (H )| − |E (H )| + |V (H )|, we have Λ(G/w) ≤ Λ(G)

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By applying neighborhood compression repeatedly a coloring G → K n . Since ¡ ¢ ¡ one ¢ ¡obtains ¢ n ≥ χ(G) by definition and the function n 7→ n3 − n2 + n1 is non-decreasing, we have the following corollary Corollary 4. Let G = (V, E ) be a connected graph. Then we have à ! à ! à ! χ(G) χ(G) χ(G) − + ≤ |C (G)| − |E (G)| + |V (G)| 3 2 1

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As an application, our results on edge contractions and induced non-separating cycles give a new lower bound on the number of induced non-separating cycles. The best-known lower bound of such comes from the fact that the cycle space of a 3-connected graph is generated by the induced non-separating cycles. Namely, let G = (V, E ) be a connected simple graph. The edge space, denoted E (G), is a free Z2 -module with basis E . An element of E (G) is called a cycle if it is a sum of indicators of edges of a cycle in G. The submodule of E (G) generated by the cycles is called the cycle space of G and denoted by C (G). The rank of C (G) is equal to the rank of the fundamental group of G, which equals to the number of edges that lies outside of a spanning forrest in G. Suppose G is connected, and let T be a spanning tree of G. Then |V (T )| = |V (G)| and |E (T )| = |V (G)| − 1, so we have rank(C (G)) = |E (G)| − |E (T )| = |E (G)| − |V (G)| + 1.

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It is easy to see that C (G) is generated by the set of induced cycles. If G is planar, we only need “facial” cycles to generate the cycle space. Induced non-separating cycles generalize this notion of facial cycles for planar graphs to a general 3-connected graphs, and Tutte [6] showed that these cycles indeed generate the cycle space for 3-connected graphs. This gives the following best-known lower bound of |F (G)|: |F (G)| ≥ |E (G)| − |E (T )| = |E (G)| − |V (G)| + 1.

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In fact, this is a special case of the inequality given in Corollary 2, which says that a graph with large clique minor contains many induced non-separating cycles. Graphs with many edges are known to have large Hadwiger number by Kostochka [2]. Namely, for graphs G = (V, E ) p with |E | = k|V | for a fixed integer κ ≥ 2, we have h(G) ≥ k/270 log k. Noting that the left hand side of (3) is non-decreasing in h(G), Corollary 2 yields the following new lower bound on |F (G)|: Corollary 5. Let G = (V, E ) be a 3-connected graph with |E |/|V | = k for some integer k ≥ 2. Then we have |F (G)| ≥ (k − 1)|V | +

¡ 3 ¢ 1 k log3/2 k − 1620k 2 log k + 801900k log1/2 k . 118098000 3

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2. Proof of Theorem 1 The following lemma describes how induced non-separating cycles behave when contracting an edge e such that the resulting graph is still 3-connected. Lemma 2.1. Let G = (V, E ) be a 3-connected graph and e = uv be an edge such that G/e is 3-connected. Then there is an injection ψ : F (G/e) → F (G) such that ψ(C )/e = C for all C ∈ F (G/e). Furthermore, we have Te (G) ⊂ F (G) \ im(ψ), where Te (G) is the set of all triangles in G using the edge e and im(ψ) = ψ[F (G/e)]. Proof. Denote the set of all subgraphs of G by P (G). Recall that contracting the edge e = uv is the operation of identifying the vertices u and v into a single vertex, say v e . Such an operation gives a map c e : P (G) → P (G/e) by H 7→ H /e. Note that for any H ⊂ G/e, we have G/e − H = G − c e−1 (H ). We first construct an injection ψ : C (G/e) → C (G) such that ψ(C )/e = C . For each C ∈ F (G/e), we define   if v e ∉ V (C ) C     if degce−1 (C ) (u) = 3 and degce−1 (C ) (v) < 3  (C − v e ) + u ψ(C ) =

(C − v e ) + v     (C − v e ) + u + v   (C − v ) + u or (C − v ) + v e e

if degce−1 (C ) (u) < 3 and degce−1 (C ) (v) = 3 if degce−1 (C ) (u) < 3 and degce−1 (C ) (v) < 3 if degce−1 (C ) (u) = 3 and degce−1 (C ) (v) = 3

Let C ∈ F (G/e), and suppose C uses the contracted vertex v e . Let x, y be the two neighbor of v e in C . Then we have c e−1 (C ) = (C − v e ) + u + v. Since v e is in C , both x and y must be adjacent to either u or v. Hence c e−1 (C ) contains a cycle in G containing the induced path C − v e and the edge e. On the other hand, since contracting e in c e−1 (C ) yields the induced cycle C in G/e, both u and v cannot be adjacent to any vertices of C − v e other hand x and y. So the five cases in the above definition exhaust all possibilities, where the last four of which are illustrated in Figure 2. It is easy to check that ψ is a map from C (G/e) to C (G). Since C − v e ⊂ ψ(C ) ⊂ c e−1 (C ), we also have ψ(C )/e = C for each C ∈ C (G/e). This implies the injectivity of ψ. Also, the second part of the assertion follows from this property.

Figure 1: Possible types of inverse image of an induced cycle in G/e using the contracted vertex v e under the contraction G → G/e.

Now we wish to show that ψ restricts to a map F (G/e) → F (G). That is, we want to show that for each C ∈ C (G/e), if C is non-separating in G/e, then ψ(C ) is also non-separating in G. Suppose C ∈ F (G/e). First assume v e is not in C . Then c e−1 (C ) = C , so G − C = G − c e−1 (C ) = G/e −C is connected. Hence C ∈ F (G). Now we may assume C uses v e . Note that in the fourth case, c e−1 (C ) = ψ(C ) so that G − ψ(C ) = G/e − C , so ψ(C ) must be non-separating. Suppose degce−1 (C ) (v) < 3. Since G is 3-connected, v must be adjacent to a component of G − c e−1 (C ). Hence the singleton v does not form a component by itself in G − ψ(C ). But since G − c e−1 (C ) has only one component and c e−1 (C ) = ψ(C ) + v, we have that G − ψ(C ) also has only one component. So ψ(C ) ∈ F (G). The third case is symmetric to the previous case, so it remains to check the last case. 4

Suppose the last of the five cases above. We wish to show that either (C − v e ) + u or (C − v e ) + v is non-separating. Suppose for contrary that both of them are separating. Let B = G −c e−1 (C ) ⊂ G. If u is adjacent to B in G, then (C −v e )+v is non-separating. So we may assume u is not adjacent to B . Similarly, we may assume v is not adjacent to B . Let L = u + v ⊂ G. Then {x, y} is a 2-separation of G that separates L and B , both of which are non-null. But this contradicts the 3-connection of G. This shows that ψ restricts to a map F (G/e) → F (G). Proof of Theorem 1. Let G = (V, E ) be a 3-connected graph and let e ∈ E be such that G/e remains 3-connected. Let Te (G) be the set of all triangles in G using the edge e as in Lemma 2.1. It is easy to observe that |E (G)| − |E (G/e)| = |Te (G)| + 1. On the other hand, the second part of Lemma 2.1 gives us |F (G)| − |im ψ| ≥ |Te (G)|. Combining the above inequalities then gives Λ(G) − Λ(G/e) = (|F (G)| − |im ψ|) − (|Te (G)| + 1) + 1 ≥ 0 as desired. This shows the assertion. ■ In order to prove Corollary 2, we need the existence of an edge e ∈ E (G) such that G/e remains 3-connected and h(G/e) = h(G). This is provided by Seymour’s “Splitter theorem” in [4]. Theorem 2.3 (Seymour [4]). Let H be 3-connected minor of a 3-connected graph G such that |V (G)| > 4. If |V (H )| < |V (G)|, then there exists an edge uv ∈ E (G) such that G/uv is 3-connected and contains H as a minor. Proof of Corollary 2. Let G = (V, E ) be a 3-connected graph. We use induction on |V |. Let K = K h(G) be a maximal clique minor of G. Note that the assertion is equivalent to show that Λ(K h(G) ) ≤ Λ(G) ¡ ¢ ¡ ¢ ¡ ¢ since Λ(K n ) = n3 − n2 + n1 . We may assume K 6= G since there is nothing to prove otherwise. If |V | = 4, then 3-connection yields G = K 4 so the assertion holds. For the induction step, we may assume the assertion for 3-connected graphs with < |V | vertices. If |V (K )| = |V (G)|, then the complete graph K can be obtained from G only by deleting edges, a contradiction. Hence |V (K )| < |V (G)|. Then by Theorem 2.3, there exists an edge e in G such that G/e is 3-connected and still contains K as a minor. Thus K ≤m G/e ≤m G, which yields h(G/e) = h(G). Then by the induction hypothesis and Theorem 1, we obtain Λ(K h(G) ) = Λ(K h(G/e) ) ≤ Λ(G/e) ≤ Λ(G) as desired. ■ We conclude this section by giving a short proof to the following theorem of Tutte: Theorem 2.4 (Tutte [6]). Let G = (V, E ) be a 3-connected graph. Then the cycle space of G is generated by its induced non-separating cycles. 5

Proof. We may assume |V | ≥ 5 since otherwise G is a complete graph and the assertion is clear. Let F (G) be the submodule of C (G) generated by the elements of F (G). We wish to show that this submodule contains all cycles if G is 3-connected. Let G = (V, E ) be a minimal counter example. Let α be any cycle in G not in F (G). Let e be an edge in G such that G/e is 3-connected (the existence of such “contractible” edge follows from Theorem 2.3, but a short proof of which is given in Lemma 3.2.1 in [1]). We may assume that α is not a triangle using the edge e since otherwise it is non-separating by the second part of Lemma 2.1. So α/e is a cycle in G. But since F (G/e) = C (G/e) by the minimality of G and the choice of e, we have α/e ∈ F (G/e). So there are elements F 1 , · · · , F n ∈ F (G/e) such that α/e =

n X

Fi .

i =1

We may choose α so that n is as small as possible. First suppose n = 1 so that α/e = F 1 . Then there are five cases as in the definition of ψ in Lemma 2.1, and it is easy to see that α is a Z2 -sum of ψ(F 1 ) and some (or none) of triangles in c −1 (F 1 ) using edge e (for type (d) in Figure 2, choose ψ(F 1 ) appropriately), which are all members of F (G). So α ∈ F (G), contrary to our assumption. Thus we must have n > 1. But then observe that (α + ψ(F n ))/e = α/e + ψ(F n )/e =

n X

Fi + Fn =

i =1

n−1 X

Fi .

i =1

So by the minimality of α, we have α + ψ(F n ) ∈ F (G). But since ψ(F n ) ∈ F (G), we then have α = (α + ψ(F n )) + ψ(F n ) ∈ F (G), a contradiction. This shows the assertion. 3. Proof of Theorem 3 A graph G = (V, E ) is a cone graph if there exists a subgraph H ⊂ G and a vertex w such that w is adjacent to every vertex in H and G = H + w. In such case, we write G = H w and call w an apex and H a base. Recall the definition of neighborhood compression G → G/w given in the introduction, which is a homomorphism that maps the induced subgraph Γ := G[N (w)] onto a complete graph B := K m where m = χ(G[N (w)]) and leaves the subgraph H = G − N (w) − w unchanged. In other words, it “compresses” the cone graph Γw into B w via a homomorphism `3 (see Figure 2).

w

w 

1

2

3

3

2

B

1

2

3

H

H G

G/w

Figure 2: An example of neighborhood compression G → G/w. Here Γ := G[N (w)] is isomorphic to C 5 which has chromatic number 3, so after compression it becomes B ' K 3 . Corresponding 3-coloring of Γ is labeled in the figure.

We first state a simple observation about the number of induced cycles in a cone graph. 6

Proposition 3.1. Let G = H w be a cone graph. Then one has |C (H w )| = |C (H )| + |E (H )|. Proof. There are |C (H )| induced cycles in G that does not use the apex w. On the other hand, for any adjacent vertices u, v in H , C := u + v + w forms a triangle so there are at least |E (H )| many induced cycles in H w that uses the apex w. Since any induced cycle C in H w using w must use exactly two distinct vertices in H , the assertion follows. In order to prove Theorem 3, we need to understand how induced cycles behave under neighborhood compression. This is given by the following lemma. Lemma 3.2. Let G = (V, E ) be a connected graph with a vertex w. Let c be a homomorphism G → G/w and denote Γ = G[N (w)], H = G − Γ − w, and c[Γ] = B as in Figure 2. Then exists an injection φ : C (G/w) \ C (B w ) → C (G) \ C (G/w) such that φ(C )/w − w = C for all C ∈ C (G/w). Furthermore, we have |C (G) \C (Γw )| − |C (G/w) \C (B w )| ≥ |E (G) \ E (Γw )| − |E (G/w) \ E (B w )|

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Proof. Let We keep the usual notation Γ = G[N (w)], H = G − Γ − w, and c[Γ] = B . Label the vertices of B from 1 to k = |V (B )|. Let C be an induced cycle in G/w not contained in B w and we correspond an induced cycle φ(C ) in G. Observe that C does not use the apex w and contains at most two vertices in B , since otherwise it would be a triangle in B w . If C uses no vertex in B w then it is an induced cycle in G and define φ(C ) = C . Otherwise, C uses either one or two vertices in B . First suppose C uses one vertex, say z, in B . Let x, y be the two neighbors of z in C . Then P := C − z ⊂ H is an induced path from x to y both in G/w and G. If there exists a vertex v ∈ c −1 (z) such that v x, v y ∈ E (G), then define φ(C ) = P + v. If not, there exists two vertices u, v ∈ c −1 (z) such that uv, v y ∈ E (G). Note that uv ∉ E (G) since the homomorphism c identifies u and v into a single vertex z. So P +u +v +z is an induced cycle in G, and we define this to be φ(C ). On the other hand, suppose C uses two vertices, say z 1 and z 2 , in B . Let x, y be the neighbors of z 1 , z 2 in C repectively. Then P := C − z 1 − z 2 ⊂ H is an induced path from x to y in both G/w and G. Since c is a homomorphism, there exists vertices v 1 , v 2 in Γ such that c(v i ) = z i for i ∈ {1, 2} and xv 1 , y v 2 ∈ E (G). Note that v 1 y, v 2 x ∉ E (G) from construction. Now if v 1 v 2 ∈ E (G) we define φ(C ) = P + v 1 + v 2 , or otherwise φ(C ) = P + v 1 + v 2 + w. This defines the map φ : C (G/w) \ C (B w ) → C (G) \ C (G/w). The asserted property φ(C )/w − w = C in all cases is clear and the injectivity follows from this. The second part of the assertion says that we lose more induced cycles than edges by the neighborhood compression c : G → G/w outside Γw . It suffices to show that the number of induced cycles in C (G) \ C (Γw ) that are not in the image of φ is at least the right hand side of (9). Let R the set of “wedges” in G − w that becomes a K 2 by c, i.e., R = {P a path in G | V (P ) = {x, y, z}, {x, z} ⊂ V (Γ), y ∈ V (H ), and c(x) = c(y)} Note that the conditions on path P ∈ R implies that P is a length 2 path from x to z and that these two endpoints are identified by the homomorphism c. This implies zx ∉ E (G). Hence for each P ∈ R, P + w is a rectangle in G whose image under c is a length 2 path in G/w. By the property of φ, these rectangles are not in the image of φ. On the other hand, each such wedge P ∈ R corresponds to a pair of edges in G not in Γw that are identified by c. Thus |R| equals to the right hand side of (9). This shows the assertion. Proof of Theorem 3. We use induction on |V |. We may assume |V | > 1 since otherwise the assertion is trivial. We suppose the assertion for any pair of connected graph G = (V, E ) and 7

a fixed vertex w with |V | < n. For the induction step, let G = (V, E ) be any connected graph with |V | = n and fix a vertex w ∈ V . Let c : G → G/w, Γ, B, and H be as in Lemma 3.2. Let ∆C := |C (G)| − |C (G/w)| and define ∆E and ∆V similarly. We claim that |C (Γw )| − |C (B w )| ≥ |E (H )| − |E (K )| We first see how the assertion follows from this claim. Proposition 3.1 gives |E (Γw )| − |E (B w )|

=

|E (Γ)| − |E (B )| + |V (Γ)| − |V (B )|

=

|E (Γ)| − |E (B )| + ∆V

Then using the claim, Lemma 9, and the above observation, we have ¡ ¢ ∆C = |C (Γw )| − |C (B w )| + |C (G) \C (Γw )| − |C (G/w) \C (B w )| ≥

|E (Γ)| − |E (B )| + |E (G) \ E (Γw )| − |E (G/w) \ E (B w )|

=

∆E − ∆V

as desired. It remains to show the claim. Note that Proposition 3.1 yields |C (Γw )| − |C (B w )| = |C (Γ)| − |C (B )| + |E (Γ)| − |E (B )|, so it suffices to show that Γ has more induced cycles than B . To this end, we first observe that for any connected graph G 0 = (V 0 , E 0 ) and a vertex w 0 ∈ V 0 , a neighborhood compression G 0 → G 0 /w 0 decreases the number of induced cycles. Indeed, the induction hypothesis implies (keeping the similar ∆ notation for G 0 → G 0 /w 0 ) ∆C ≥ ∆E − ∆V and the right hand side of the above inequality is at least 0, since the neighborhood compression c : G → G/w identifies more edges then vertices; for, if two non-adjacent vertices u and v in Γ are identified by the neighborhood compression c, then the two edges uw and v w are also identified. Now, note that the subgraph Γ ⊂ G may not be connected; let Γ1 , · · · , Γk be its components. For each component Γi , apply a sequence of neighborhood compression until one gets a complete graph, say, B i . Since each of the component misses the vertex w, one can apply the above observation so that |C (Γi )| ≥ |C (B i )|

1≤i ≤k

Next, through a sequence of nonedge contractions, embed all complete graphs K i into a maximal one, say B ∗ . Note that |C (Γ)| =

k X

|C (Γi )| ≥ max (|C (B i )|) = |C (K ∗ )|

i =1

1≤i ≤k

On the other hand, the composition of all homomorphisms we have used gives a coloring Γ → B ∗ . But by the minimality of B ' K χ(Γ) , we have |V (B )| ≤ |V (B ∗ )| and hence |C (Γ)| ≥ |C (B )| as desired. This shows the assertion. ■ 8

Proof of Corollary 4. Let G = (V, E ) be a connected graph. By applying neighborhood compression repeatedly until one obtains a complete graph, say K n , one gets a coloring G → K n . By Theorem 4 we have Λ(K n ) ≤ Λ(G). Then the assertion follows since χ(G) ≤ n by definition and since Λ(K n ) is non-decreasing in n. ■

4. Concluding remarks In Theorem 1 we showed that the graph parameter Λ(G) = |F (G)|−|E (G)|+|V (G)| has a monotonicity property under edge contractions. However, similar monotonicity property of Λ under edge deletions is not necessarily ture, that is, we are not guaranteed to have Λ(G − e) ≤ Λ(G)

(10)

for any 3-connected graph G with G − e also 3-connected. Note that (10) is equivalent to say that one loses at least one induced non-separating cycles after an edge gets deleted. Indeed, Thomassen [5] showed that in a 3-connected graph, there are at least two induced nonseparating cycles using any fixed edge e, just like the facial cycles of a planar graph do. However, it could be that many non-separating cycles in G have “chord” e so that deletion of which makes all of them induced. More specifically, let G be a graph with a vertex w such that w is adjacent to all vertices in H := G − w, where H is a 2-connected graph consisting of n vertex-disjoint induced paths P 1 , · · · , P n from a source s ∈ V to a sink t ∈ V . Suppose that st ∈ E (G) and all of P i ’s have length ≥ 2. Note that every induced cycle in H and H −e is non-separating in G and G−e, respectively. Now H has exactly n induced cycles, whereas H − e has n(n − 1)/2 of them. Consequently, |F (G)| = n + |E (H )| but |F (G − e)| = n(n − 1)/2 + |E (H )| − 1. Nevertheless, if instead one considers a larger class of cycles which are non-separating but not necessarily induced, then the monotonicity of the corresponding graph parameter Λ under both edge contractions and deletions hold. Hence in this case we have Λ(H ) ≤ Λ(G) for any 3-connected minor H of G, not necessarily a clique. On the other hand, it is a future goal to prove Corollary 4 for a smaller class of cycles assuming certain connectivity conditions on G. This would require a “chain theorem”, which is a homomorhpism counterpart of the Splitter theorem that gives the existence of a certain homomorphism preserving a given connectivity. For example, one could try to prove Corollary 2 for the chromatic number, aiming at a short proof of the four color theorem. Indeed, Kriesell [3] showed that 3-connected graphs always have a contractible nonedge, i.e., a pair of non-adjacent vertices u, v such that identifying them yields a 3-connected graph. However, it seems that cycles behave less nicely under nonedge contractions than under edge contractions, as identifying two nonadjacent vertices could make a non-separating induced path into a non-separating induced cycle. A way around this difficulty is to view a nonedge contraction as a composition of adding a new edge and then contracting it. Namely, if we use the same notation G → G/e for contracting a nonedge e, then note that G/e = (G + e)/e. Hence if G/e is 3-connected and Λ(G + e) ≤ Λ(G), then by Theorem 1 we have Λ(G/e) = Λ[(G + e)/e] ≤ Λ(G + e) ≤ Λ(G), from which Corollary 2 for the chromatic number easily follows. Hence we ask the following question: Problem 4.1. Let G = (V, E ) be a 3-connected graph. Show that there exists a nonedge e such that G/e is 3-connected and |F (G + e)| ≤ |F (G)| + 1. 9

Acknowledgement The author give special thanks to Neil Robertson for helpful discussions and Paul Seymour for valuable suggestions. References [1] Diestel, R., 2000. Graph theory {graduate texts in mathematics; 173}. Springer-Verlag Berlin and Heidelberg GmbH & amp. [2] Kostochka, A. V., 1984. Lower bound of the hadwiger number of graphs by their average degree. Combinatorica 4 (4), 307–316. [3] Kriesell, M., 1998. Contractible non-edges in 3-connected graphs. Journal of Combinatorial Theory, Series B 74 (2), 192–201. [4] Seymour, P., 1986. Decomposition of regular matroids. A source book in matroid theory, 339. [5] Thomassen, C., 1980. Planarity and duality of finite and infinite graphs. Journal of Combinatorial Theory, Series B 29 (2), 244–271. [6] Tutte, W. T., 1963. How to draw a graph. Proc. London Math. Soc 13 (3), 743–768. [7] Vince, A., Little, C. H., 1989. Discrete jordan curve theorems. Journal of Combinatorial Theory, Series B 47 (3), 251–261. [8] Yu, X., 1992. Non-separating cycles and discrete jordan curves. Journal of Combinatorial Theory, Series B 54 (1), 142–154.

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