Polynomials counting nowhere-zero chains in graphs

We introduce polynomials counting nowhere-zero chains in graphs – nonhomogeneous analogues of nowhere-zero flows. For a graph G, an Abelian group A, and b : V (G) → A, let αG,b be a mapping from Λ(G) (a family of vertex sets of connected subgraphs of G satisfying an additional condition) to {0, 1} such that for each X ∈ Λ(G), αG,b(X) = 0 if and only if ∑ v∈X b(v) = 0. We prove that there exists a polynomial function F (G,α; k) (α = αG,b) of k such that for any Abelian group A′ of order k and each b′ : V (G)→ A′ satisfying αG,b′ = α, F (G,α; k) equals the number of nowhere-zero A′-chains φ in G having boundaries equal to b′. In particular F (G,α; k) is the flow polynomial of G if α(X) = 0 for each X ∈ Λ(G). Finally we characterize α for which F (G,α; k) is nonzero and show that in this case F (G,α; k) has the same degree as the flow polynomial of G. Mathematics Subject Classifications: 05C31, 05C21, 05C15


Introduction
The graphs considered in this paper are finite and unoriented. Multiple edges and loops are allowed. If G is a graph, then V (G) and E(G) denote the sets of vertices and edges of G, respectively. With each edge of G there are associated two distinct arcs (see [10]). Arcs on distinct edges are distinct. If an arc on an edge is denoted by x the other is denoted by x −1 . If the ends of an edge e of G are vertices u and v, one of the arcs on e is said to be directed from u to v and the other one is directed from v to u. Let D(G) denote the set of arcs on G. If v ∈ V (G) (X ⊆ V (G)), then the set of arcs of G directed out from v (resp. X) is denoted by ω G (v) (resp. ω G (X)).
If G is a graph and A is an Abelian group, then a nowhere-zero A-chain in G is a mapping ϕ from D(G) to A \ {0} such that ϕ(x −1 ) = −ϕ(x) for every x ∈ D(G). By the boundary of ϕ we mean the mapping ∂ϕ from V (G) to A such that ∂ϕ(v) = x∈ω G (v) ϕ(x) for every v ∈ V (G). By an easy computation (see [ In particular, if |ω G (X)| = 0 then v∈X ∂ϕ(v) = 0 and if |ω G (X)| = 1 then v∈X ∂ϕ(v) = 0. If ∂ϕ(v) = 0 for every v ∈ V (G), then we call ϕ a nowhere-zero A-flow.
Nowhere-zero flows in graphs have been introduced by Tutte [15] as dual concept to graph coloring problems. It is well known that a planar graph is k-colorable if and only if its dual admits a nowhere-zero A-flow for any Abelian group of order k. This paper is motivated by the facts that nowhere-zero flows are often studied in the framework of nowhere-zero chains (see cf. [8,9,12,13]) and that the number of nowherezero A-flows in a graph is evaluated by a polynomial function of |A|, the flow polynomial (see cf. [1,2,3,4,7,14]). Notice that flow polynomials are studied in framework of other combinatorial invariants, in particular the Tutte and characteristic polynomials -topics having applications not only in combinatorics, but also in coding theory, geometry, knot theory and physics (see [6,5,16,17]).
It is natural to ask whether there exist polynomial functions of |A| that generalize flow polynomials and count nowhere-zero A-chains in graphs (but we are not aware whether this question appeared somewhere in the literature). If a graph consists from an isolated vertex v and a loop, then the number of nowhere-zero A-chains ϕ satisfying ∂ϕ . With respect to similar arguments we need to consider so called "{0, 1}-assignings" of certain sets of vertices and in the main result (Theorem 1) we show that the number of nowhere zero A-chains with boundaries corresponding to an assigning is evaluated by a polynomial function of |A| that we call an assigning polynomial. As a consequence we can guarantee existence of group-valued chains if there exist chains with the same assigning and having values from a group of the same cardinality (Corollary 2). In the third section we decompose assigning polynomials and characterize assignings corresponding to nonzero polynomials. In the last section we present several examples and remarks about the topic.

Assigning polynomials
In this section we introduce the polynomial counting nowhere-zero chains in graphs.
For a graph G, let c(G) denote the number of components of G and m(G) = |E(G)| − |V (G)| + c(G). Denote by Λ(G) the family of X ⊆ V (G) such that G[X] (the subgraph of G induced by X) is either a component of G or G[X] is connected and c(G − X) = c(G). Notice that ∅ ∈ Λ(G). By an assigning of G we mean any mapping α from Λ(G) to {0, 1}. We write α ≡ 0 if α(X) = 0 for each X ∈ Λ(G).
Let b : V (G) → A. By α G,b we denote the assigning of G such that for each X ∈ Λ(G), α G,b (X) = 0 if v∈X b(v) = 0 and α G,b (X) = 1 otherwise. We also say that α is called G-proper and α = α G,b is called a G-proper assigning of G. Clearly, each G-proper assigning is G-admissible, but not vice versa (more details are presented in the last section and Theorem 4).
For any edge e of G, G/e and G−e denote the graph arising from G after contracting and deleting e, respectively. Let e ∈ E(G) have ends v 1 , v 2 . Denote by v /e the vertex of G/e arising from v 1 and v 2 after contracting e. For each X ∈ Λ(G/e) denote by X e = (X \ {v /e }) ∪ {v 1 , v 2 } if v /e ∈ X and X e = X otherwise. Clearly, X ∈ Λ(G/e) if and only if X e ∈ Λ(G). Thus for any assigning α of G we can define α /e to be the assigning of G/e such that α /e (X) = α(X e ) for each X ∈ Λ(G/e). Moreover Λ(G−e) ⊆ Λ(G). Thus for any assigning α of G we can define α −e to be the assigning of G−e such that α −e (X) = α(X) for each X ∈ Λ(G−e). Clearly if e is a loop (i.e, v 1 = v 2 ), then v /e = v 1 = v 2 , Λ(G) = Λ(G/e) = Λ(G−e), and α = α /e = α −e . Theorem 1. Suppose that α is a G-admissible assigning of a graph G. Then there exists a polynomial function F (G, α; k) of k such that F (G, α; k) = |Φ G (b)| for every Abelian group A of order k and every mapping b : Proof Let e be an edge of G having different ends v 1 and v 2 . Since α is G-admissible, there exist an Abelian group A and b : Thus α /e and α −e are (G/e)-and (G−e)-admissible, respectively.
Each ϕ ∈ Φ G/e (b) can be considered as a nowhere-zero A-chain ϕ in G−e (considering D(G/e) and D(G−e) to be identical). Applying (1) for ϕ and ϕ , we get that ∂ϕ Let α be G-proper. Then by (3) also α /e is (G/e)-proper. By induction hypothesis and . If e is not a bridge in G, then m(G) = m(G/e) = m(G−e) + 1 whence F G,α (k) has degree m(G). Let e be a bridge in G. Then m(G) = m(G/e) = m(G−e). Let H be the component of G containing e and H 1 , H 2 be the components of H − e. Using (1) and the fact that α is G-proper we have α(V (H)) = 0 and α(V (H 1 )) = α(V (H 2 )) = 1, whence by definition of α −e , α −e (V (H 1 )) = α −e (V (H 2 )) = 1. Therefore by (1), α −e is not (G−e)-proper and by the induction hypothesis, If α is not G-proper, then by induction hypothesis and (3), Corollary 2. Suppose that G is a graph, A, A are Abelian groups of the same order and mappings b : Corollary 2 generalizes the well known property of nowhere-zero flows, namely that the number of nowhere-zero A-flows in a graph does not depend on the structure of A but only on |A|.
The situation in Theorem 1 and Corollary 2 could be much simpler if we replace Λ(G) by the powerset of V (G). But in the last section we present an example of b, b : V (G) → A such that α G,b equals α G,b if these are considered as mappings form Λ(G), but the equality does not hold if α G,b and α G,b are considered as mappings from 2 V (G) . Thus we get |Φ G (b)| = |Φ G (b )| by Corollary 2, which is not possible if we consider 2 V (G) instead of Λ(G). For this reason we need to consider Λ(G) as small as possible, but, in the same time, Λ(G) must be sufficiently rich so that we are able to prove Theorem 1. Thus we choose a few technical definition of Λ(G).

Properties of assigning polynomials
In this section we present a decomposition formula for assigning polynomials. Using the formula we characterize assignings corresponding to nonzero polynomials.
Suppose that α is a G-admissible assigning of a graph G. Then there exists an Abelian group A and b : V (G) → A such that α G,b = α. By a b-extensionα G,b (simply an extensionα) of α we mean a mapping from 2 V (G) to {0, 1} such that for each X ⊆ V (G), α G,b (X) =α(X) = 0 if and only if v∈X b(v) = 0. (In the following section we present an example when α G,b = α G,b andα G,b =α G,b , i.e., α does not need to have a unique extension.) Let < be a linear ordering of E(G). We say that Y ⊆ E(G) is <-compatible if for each circuit C of G, C ∩ Y = {min(C)} (considering circuits as edge sets of 2-regular connected subgraphs of G). Denote by E(G, <) the set of all <-compatible subsets of E(G). Clearly, no <-compatible set contains a loop (a circuit consisting of a single edge).
Theorem 3. Suppose that α is a G-admissible assigning of a graph G,α is an extension of α, and let < be a linear ordering of E(G). Then Proof. We prove (5) Let Y ∈ E(G/e, <) and C be a circuit in G. If e is a chord of C, then from C arise two circuits C 1 , If e is not a chord of C, then from C arise just one circuit C in G/e and Y ∩ C = {min(C )}, min(C) = min(C ), whence Y ∩ C = {min(C)}. Thus Y ∈ E(G, <) and e / ∈ Y , i.e., E(G/e, <) ⊆ E + . Let Y ∈ E + and C be a circuit in G/e. Then C indicates just one circuit C e in G (where either C = C e , or C e = C ∪ {e}). Since Y ∩ C e = {min(C e )} and e / ∈ Y , we have Y ∩ C = {min(C)}. Thus E + ⊆ E(G/e, <) and E + = E(G/e, <).
It is an easy exercise to check that if Y ∈ E(G/e, <) = E + , then m( Thus by (2) and the induction hypothesis, concluding the proof.
Notice that the summands from the right hand side of (5) can differ depending on the choice ofα and <, but their sum is always F (G, α; k). Clearly, ∅ ∈ E(G, <) and by (a), δ ∅,α = 1. Thus the right hand side of (5) is a sum of powers of (k − 1) such that exactly one of them (corresponding to Y = ∅) has the maximal possible degree m(G). Hence F (G, α; k) = 0 and α is G-proper.
We show that if α ≡ 0, then F (G, α; k) is the flow polynomial of graphs (see cf. [1,2,3,14]). Proof. Clearly, α ≡ 0 is G-admissible and by Theorem 4, it is G-proper if and only if G is bridgeless. Thus the statement follows from Theorem 1 after setting α ≡ 0.

Concluding remarks
. Thus α C,b = α C,b andα C,b =α C,b . By Theorem 1 we have |Φ G (b)| = |Φ G (b )|, but we cannot apply this implication if we consider α as mappings from 2 V (G) in Theorem 1. In fact we can check that for each X ∈ Λ c (G), v∈X b(v) = 0 if and only if v∈X b (v) = 0. Thus we get |Φ G (b)| = |Φ G (b )| if we consider α as mappings from Λ c (G) in Theorem 1.
In the following example we show that assumption α G,b = α G,b cannot be deleted from Corollary 2. In fact we show that F (G, α; k) can differ from F (G, α ; k) if α(X) = α (X) for a unique X ∈ Λ(G) (which, furthermore, has cardinality > 1). Example 5. Let C be a circuit of order 6 with consecutive vertices v 1 , . . . , v 6 and b, b : it is easy to check that X and V (C) \ X are the unique elements of Λ(C) such that α C,b (X) = α C,b (X)). Let x i denote the arc directed from v i to v i+1 (considering the indices mod 6). Let ϕ ∈ Φ C (b). If ϕ(x 1 ) = 2, then we can check that (ϕ(x 1 ), . . . , ϕ(x 6 )) = (2, 3, 4, 3, 2, 1) and ϕ(x 1 ) cannot equal 3 (resp. 4, or 1) because then ϕ(x 3 ) (resp. ϕ(x 2 ), or ϕ(x 6 )) equals 0. Thus |Φ C (b)| = 1. Similarly if ϕ ∈ Φ C (b ) and ϕ (x 1 ) = 2 (resp. ϕ (x 1 ) = 3), then (ϕ (x 1 ), . . . , ϕ (x 6 )) = (2, 3, 1, 3, 2, 1) (resp. (ϕ (x 1 ), . . . , ϕ (x 6 )) = (3, 4, 2, 4, 3, 2))) and ϕ (x 1 ) cannot equal 4 (resp. 1), because then ϕ (x 2 ) (resp. ϕ (x 6 )) equals 0. Hence |Φ C (b )| = 2, i.e., |Φ C (b)| = |Φ C (b )|. On the other hand considering b as defined in Example 4, we have proved that By a leaf of a forest T we mean any vertex of degree 1 or 0. For each component H of Assume that G is a graph where V (G) = {v 1 , . . . , v n } has cardinality n 1 and let T be a forest having n leaves v 1 , . . . , v n . For i = 1, . . . , n, identify v i with v i into a new vertex v i . Denote the resulting graph by G T . Let A be an Abelian group. We say that a nowhere-zero A-chain ϕ in G is a nowhere-zero (A, T )-chain if v∈X ∂ϕ(v) = 0 for each X ∈ L(T ) and v∈Y ∂ϕ(v) = 0 for each edge e of T and Y ∈ L(e). It is easy to check that if ϕ is a nowhere-zero A-flow in G T then the restriction ψ of ϕ into D(G) is a nowhere-zero (A, T )-chain in G and if ψ is a nowhere-zero (A, T )-chain in G then it can be extended into a unique nowhere-zero A-flow in G T . Thus there is a one-to-one correspondence between the sets of nowhere-zero A-flows in G T and nowhere-zero (A, T )-chains in G. Therefore the flow polynomial F (G T ; k) of G T counts the number of nowhere-zero (A, T )-chains in G for k = |A|.  3 , v 4 }} (T consists of three copies of K 2 ). This example shows that the notion of (A, T )-mapping does not have an analogous application as presented in Corollary 2.
Suppose that T is a forest such that exactly one component of T is isomorphic either with K 2 , or with K 1,3 , and all other components are isolated vertices. It is easy to check that for each (A, T )-mapping b on V (G) we have F (G, α G,b ; k) = F (G T ; k). But in general we do not know whether for any assigning α of a graph G, the polynomial F (G, α; k) can be evaluated by polynomials F (G T ; k) for some forests T .