Permanent index of matrices associated with graphs

A total weighting of a graph $G$ is a mapping $f$ which assigns to each element $z \in V(G) \cup E(G)$ a real number $f(z)$ as its weight. The vertex sum of $v$ with respect to $f$ is $\phi_f(v)=\sum_{e \in E(v)}f(e)+f(v)$. A total weighting is proper if $\phi_f(u) \ne \phi_f(v)$ for any edge $uv$ of $G$. A $(k,k')$-list assignment is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ permissible weights, and assigns to each edge $e$ a set $L(e)$ of $k'$ permissible weights. We say $G$ is $(k,k')$-choosable if for any $(k,k')$-list assignment $L$, there is a proper total weighting $f$ of $G$ with $f(z) \in L(z)$ for each $z \in V(G) \cup E(G)$. It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph is $(2,2)$-choosable and every graph with no isolated edge is $(1,3)$-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. This approach leads to conjectures on the permanent indices of matrices $A_G$ and $B_G$ associated to a graph $G$. In this paper, we establish a method that reduces the study of permanent of matrices associated to a graph $G$ to the study of permanent of matrices associated to induced subgraphs of $G$. Using this reduction method, we show that if $G$ is a subcubic graph, or a $2$-tree, or a Halin graph, or a grid, then $A_G$ has permanent index $1$. As a consequence, these graphs are $(2,2)$-choosable. \end{abstract} {\small \noindent{{\bf Key words: } Permanent index, matrix, total weighting}


Introduction
A total weighting of a graph G is a mapping f which assigns to each element z ∈ V (G)∪E(G) a real number f (z) as its weight. Given a total weighting f of G, for a vertex v of G, the vertex sum of v with respect to f is defined as φ f (v) = e∈E(v) f (e)+f (v). A total weighting is proper if φ f is a proper colouring of G, i.e., for any edge uv of G, φ f (u) = φ f (v). A total weighting φ with φ(v) = 0 for all vertices v is also called an edge weighting. A proper edge weighting φ with φ(e) ∈ {1, 2, . . . , k} for all edges e is called a vertex colouring kedge weighting of G. Karonski, Luczak and Thomason [7] first studied edge weighting of graphs. They conjectured that every graph with no isolated edges has a vertex colouring 3-edge weighting. This conjecture received considerable attention, and is called the 1-2-3 conjecture. Addario-Berry, Dalal, McDiarmid, Reed and Thomason [2] proved that every graph with no isolated edges has a vertex colouring k-edge weighting for k = 30. The bound k was improved to k = 16 by Addario-Berry, Dalal and Reed in [1] and to k = 13 by Wang and Yu in [10], and to k = 5 by Kalkowski [8].
Total weighting of graphs was first studied by Przyby lo and Woźniak in [11], where they defined τ (G) to be the least integer k such that G has a proper total weighting φ with φ(z) ∈ {1, 2, . . . , k} for z ∈ V (G) ∪ E(G). They proved that τ (G) ≤ 11 for all graphs G, and conjectured that τ (G) = 2 for all graphs G. This conjecture is called the 1-2 conjecture. A breakthrough on 1-2 conjecture was obtained by Kalkowski, Karoński and Pfender in [9], where it was proved that every graph G has a proper total weighting φ with φ(v) ∈ {1, 2} for v ∈ V (G) and φ(e) ∈ {1, 2, 3} for e ∈ E(G).
The list version of edge weighting of graphs was introduced by Bartnicki, Grytczuk and Niwczyk in [6], and the list version of total weighting of graphs was introduced independently by Wong and Zhu in [13] and by Przyby lo and Woźniak [12]. Suppose ψ : V (G) ∪ E(G) → {1, 2, . . . , } is a mapping which assigns to each vertex and each edge of G a positive integer. A ψ-list assignment of G is a mapping L which assigns to z ∈ V (G) ∪ E(G) a set L(z) of ψ(z) real numbers. Given a total list assignment L, a proper L-total weighting is a proper total weighting φ with φ(z) ∈ L(z) for all z ∈ V (G) ∪ E(G). We say G is total weight ψ-choosable if for any ψ-list assignment L, there is a proper L-total weighting of G. We say G is (k, k ′ )-choosable if G is ψ-total weight choosable, where ψ(v) = k for v ∈ V (G) and ψ(e) = k ′ for e ∈ E(G).
As strengthening of the 1-2-3 conjecture and the 1-2 conjecture, it was conjectured in [13] that every graph with no isolated edges is (1, 3)-choosable and every graph is (2, 2)choosable. Thes two conjectures received a lot of attention and are verified for some special classes of graphs. In particular, it was shown in [14] that every graph is (2, 3)-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. For each z ∈ V (G) ∪ E(G), let x z be a variable associated to z. Fix an orientation D of G. Consider the polynomial Assign a real number φ(z) to the variable x z , and view φ(z) as the weight of z. Let P G (φ) be the evaluation of the polynomial at x z = φ(z). Then φ is a proper total weighting of G if and only if P G (φ) = 0. Note that P G has degree |E(G)|.
An index function of G is a mapping η which assigns to each vertex or edge z of G a non-negative integer η(z) and an index function For a valid index function η, let c η be the coefficient of the monomial z∈V ∪E x η(z) z in the expansion of P G . It follows from Combinatorial Nullstellensatz [3,5] that if c η = 0, and L is a list assignment which assigns to each z ∈ V (G) ∪ E(G) a set L(z) of η(z) + 1 real numbers, then there exists a mapping φ with φ(z) ∈ L(z) such that It is straightforward to verify that for e ∈ E(G) and otherwise. Now A G is a matrix, whose rows are indexed by the edges of G and the columns are indexed by edges and vertices of G. Let B G be the submatrix of A G consisting of those columns of A G indexed by edges. It turns out that (k, k ′ )-choosability of a graph G is related to the permanent indices of A G and B G .
For an m × m matrix A (whose entries are reals), the permanent of A is defined as where S m is the symmetric group of order m, i.e., the summation is taken over all the permutations σ over {1, 2, . . . , m}. The permanent index of a matrix A, denoted by pind(A), is the minimum integer k such that there is a matrix A ′ such that per(A ′ ) = 0, each column of A ′ is a column of A and each column of A occurs in A ′ at most k times (if such an integer k does not exist, then pind(A) = ∞).
Consider the matrix A G defined above. Given a vertex or edge z of G, let A G (z) be the column of A G indexed by z. For an index function η of G, let A G (η) be the matrix, each of its column is a column of A G , and each column A G (z) of A G occurs η(z) times as a column of A G (η). It is known [4,13] and easy to verify that for a valid index function η of G, c η = 0 if and only if per(A G (η)) = 0. Thus if pind(A G ) = 1, then G is (2, 2)-choosable; if pind(B G ) ≤ 2, then G is (1, 3)-choosable. The following two conjectures are proposed in [13]: Conjecture 1 [6] For any graph G with no isolated edges, pind(B G ) ≤ 2.
We say an index function η is non-singular if there is a valid index function η ′ ≤ η with per(A G (η ′ )) = 0. In this paper, we are interested in non-singularity of index functions η for which η(e) = 1 for every edge e and η(v) can be any non-negative integers for any every vertex v. Assume η is such an index function of G. We delete a vertex v, and construct an index function η ′ for G − v from the restriction of η to G − v by doing the following modification: η(v) of the neighbours u of v have η ′ (u) = η(u)+ 1, and all the other Applying this reduction method, we prove that Conjecture 2 holds for subcubic graphs, 2-trees, Halin graphs and grids. Consequently, subcubic graphs, 2-trees, Halin graphs and grids are (2, 2)-choosable.

Reduction to induced subgraphs
To study non-singularity of index functions of G, we shall consider matrices whose columns are linear combinations of columns of A G . Assume A is a square matrix whose columns are linear combinations of columns of A G . Define an index function η A : V (G) ∪ E(G) → {0, 1, . . . , } as follows: It is known [13] that columns of A G are not linearly independent. In particular, if e = uv is an edge of G, then Thus a column of A may have different ways to be expressed as linear combinations of columns of A G . So the index function η A is not uniquely determined by A. Instead, it is determined by the way we choose to express the columns of A as linear combinations of columns of A G . For simplicity, we use the notation η A , however, whenever the function η A is used, an explicit expression of the columns of A as linear combinations of columns of A G is given, and we refer to that specific expression.
It is well-known (and follows easily from the definition) that the permanent of a matrix is multi-linear on its column vectors and row vectors: If a column C of A is a linear combination of two columns vectors C = αC ′ + βC ′′ , and A ′ (respectively, A ′′ ) is obtained from A by replacing the column C with C ′ (respectively, with C ′′ ), then By using (2) repeatedly, one can find matrices A 1 , A 2 , . . . , A q and real numbers a 1 , a 2 , . . . , a q such that where each A j is a square matrix consisting of columns of A G , with each column A G (z) appears at most η(z) times. Thus if per(A) = 0, then one of the per(A j ) = 0. Thus if per(A) = 0, then η A is a non-singular index function of G.
Theorem 1 follows from the following more general statement.
If η ′ is a non-singular index function for G ′ , then η is a non-singular index function for G.
By viewing each vertex and each edge of G ′ as a vertex and an edge of G, A G (η ′′ ) is an m × m ′ matrix, consisting m ′ columns of A G . First we extend A G (η ′′ ) into an m × m matrix A by adding k copies of the column A G (v). The added k columns has k rows (the rows indexed by edges incident to v) that are all 1's (with all these edges oriented towards v), and all the other entries of these k columns are 0. Therefore per(M ) = per(A G ′ (η ′′ ))k!, and hence per(M ) = 0.
Starting from the matrix M , for each i ∈ {1, 2, . . . , k} \ J, remove min{η(e i ), η ′′ (v i )} copies of the column A G (v i ) and add min{η(e i ), η ′′ (v i )} copies of the column A G (e i ). Denote by M ′ the resulting matrix. Proof. Since by (1) Then we expand the permanent using its multilinear property (i.e. using (2) (1)). Now each column of M ′ is a linear combination of columns of A G .
We shall show that, with the linear combination of columns of M ′ given in the above As per(M ′ ) = 0, we conclude that η is a non-singular index function for G. This completes the proof of Theorem 2. Theorem 1 follows from Theorem 2 by choosing k i = 1 and |J| = d(v) − η(v). By definition, if η ′′ is non-singular and η ′ ≥ η ′′ , then η ′ is also non-singular. So the following is equivalent to Theorem 1. We shall apply Theorem 3 repeatedly and delete a sequence of vertices in order. We need to record which vertices are deleted, and when a vertex is deleted, for which neighbours u we have η ′ (u) = η(u) + 1. For this purpose, instead of really removing the deleted vertices, we indicate the deletion of v by orient all the edges incident to v from v to its neighbours, and then choose a subset of these oriented edges (to indicate those neighbours u for which η ′ (u) = η(u) + 1).
The index function η is changing in the process of the deletion. For convenience, we denote by η i the index function after the deletion of the ith vertex. In particular, η 0 = η.
Assume a vertex v is deleted in the ith step, for each neighbour u of v (at the time v is deleted), orient the edge as an arc from v to u. After a sequence of vertices are deleted, we obtain a digraph D formed by edges incident to the "deleted" vertices. Let D ′ be the sub-digraph of@ D formed by those arcs (v, u) with u be the neighbour of v (at the time v is deleted) and for which we have η ′ (u) = η(u) + 1.
If u is deleted in the ith step, then d + D ′ (u) ≤ η i−1 (u). After the ith step, all edges incident to u are oriented. On the other hand, d − D ′ (u) is the number of indices j < i for which η j (u) = η j−1 (u) + 1, and d − The following corollary summarize the final effect of the repeated application of Theorem 1.

Corollary 1 Suppose G is a graph, η is an index function of G with η(e) = 1 for all edges e, and X is a subset of
Let η ′ be the index function defined as η ′ (e) = 1 for every edge e of G[X] and η If η ′ is a non-singular index function for G[X], then η is a non-singular index function for G.
Proof. Assume η ′ is non-singular for G[X]. We shall prove that η is non-singular for G. We prove this by induction on |V − X|. If V − X = ∅, then η = η ′ and there is nothing to prove.
Assume V − X = ∅. Since the orientation D is acyclic, there is a source vertex v / ∈ X. Let e 1 , e 2 , . . . , e k be the set of edges incident to v and e i = vv i . Therefore, by induction hypothesis, η ′′ is non-singular for G − v. Apply Theorem 1 to η ′′ and η, with J = {i : 1 ≤ i ≤ k, e i / ∈ D ′ } and k i = 1 for i ∈ J, we conclude that η is non-singular for G.

Application of the reduction method
Lemma 1 Suppose G is a connected graph, and η is an index function with η(e) = 1 for all e ∈ E(G). Assume one of the following holds:

Then η is a non-singular index function of G.
Proof. Assume the lemma is not true and G is a counterexample with minimum number of vertices.
Assume first that η(v) ≥ max{1, d G (v) − 2} for all v. By reducing the value of η if needed, we may assume that η(v) = max{1, d G (v) − 2} . Let v be a non-cut vertex of G and let v 1 , . . . , v k be the neighbours of v. Consider the graph G − v. Let η ′ be the index As G − v is connected, the condition of the lemma is satisfied by G − v and η ′ . By the minimality of G, η ′ is a non-singular index function for G − v. By Theorem 1, η is a non-singular index function for G.
Assume each vertex u has η(u) ≥ d G (u)−2 and one vertex v has η(v) ≥ d G (v). Let η ′ be the index function of G − v defined as η ′ = η except that η ′ (u) = η(u) + 1 for all neighbours u of v. Note that for all the neighbours u of v, η ′ (u) ≥ d G−v (u). Thus each component of G − v, together with η ′ , satisfies the condition of the lemma. By the minimality of G, η ′ is a non-singular index function for G − v. Apply Theorem 1 again, we conclude that η is a non-singular index function for G.
A graph G is called subcubic if G has maximum degree at most 3.
Proof. If G has maximum degree at most 3, then it follows from Lemma 1 that η(z) = 1 for all z ∈ V (G) ∪ E(G) is a non-singular index function.
A graph G is a 2-tree if there is an acyclic orientation of G (also denoted by G) such that the following hold: (1) there are two adjacent vertices v 0 , v 1 with d + G (v i ) = i (i = 0, 1). (2) every other vertex v has d + G (v) = 2, and the two out-neighbours of v are adjacent. If N + G (v) = {u, w} and (u, w) is an arc, then v is called a son of the arc e = (u, w). For an acyclic oriented graph G, for v ∈ V (G), let ρ G (v) be the length of the longest directed path ending at v. So if v is a source, then ρ G (v) = 0.
Theorem 4 Let G be a 2-tree and let η be an index function of G. Assume η(z) ≥ 1 for all z ∈ E(G) ∪ V (G), except that possibly there is one arc (u, w) with ρ G (u) ≤ 1, for which η(w) ≥ 0 and η(u) ≥ 2. Then η is non-singular for G.
Proof. Assume the theorem is not true and G is a counterexample with minimum number of vertices. If the special arc (u, w) specified in the theorem does not exist, then let e = (u, w) be an arc which has at least one son, and with ρ G (u) = 1. Note that all the sons of e are sources. Let v be a son of (u, w) and let η ′ be the index function of G ′ = G−v which is equal to η, except that η ′ (u) = η(u) + 1 ≥ 2 and η ′ (w) = η(w) − 1 ≥ 0. Then G ′ and η ′ satisfying the condition of the theorem, with e be the special edge (note that ρ G−v (u) ≤ ρ G (u) = 1). Hence η ′ is non-singular for G ′ . It follows from Theorem 1 that η is non-singular for G.
Assume the special arc e = (u, w) exists. If u is a source, then delete u, and let η ′ be the index function of G ′ = G − u which is equal to η, except that η ′ (v) = η(v) + 1 for neighbours v of u. Then η ′ (v) ≥ 1 for each vertex of G ′ , hence G ′ and η ′ satisfying the condition of the theorem. So η ′ is non-singular for G ′ , and it follows from Theorem 1 that η is non-singular for G.
If u is not a source vertex and e has a son v, then v is a source vertex. We delete v and let η ′ be the index function of G ′ = G − v which is equal to η, except that η ′ (u) = η(u) − 1 and η ′ (w) = η(w) + 1. Then G ′ and η ′ satisfying the condition of the theorem, and hence η ′ is non-singular for G ′ . It follows from Theorem 1 that η is non-singular for G.
If u is not a source vertex and e has no son, then there is an arc e ′ = (u, w ′ ) which has a son a. Since ρ G (u) ≤ 1, all the sons of e ′ are sources. If e ′ has more than one son, say a, b are both sons of e ′ , then let η ′ be the restriction of η to G − {a, b}. By the minimality of G, η ′ is non-singular for G − {a, b}. By Corollary 1 (with D consists of the four arcs incident to a, b and D ′ consists of arcs au, bw ′ ), η is non-singular for G. Assume e ′ has only one son a. Let η ′ be the restriction of η to G − {a, u}, except that η ′ (w) = 1. By the minimality of G, η ′ is non-singular for G − {a, u}. By Corollary 1 (with D consists of the four arcs incident to a, u and D ′ consists of arcs aw ′ , uw), η is non-singular for G.
Proof. First we construct an acyclic orientation of G as follows: We choose a non-leaf vertex u of T as the root of T . Orient the edges of the tree from father to son. Then orient the added edges from v i to v i+1 for i = 1, 2, . . . , n − 1, and orient the edge v 1 v n from v 1 to v n . The resulting digraph is D. Now we choose a sub-digraph D ′ of D as follows: D ′ consists of a directed path P from the root vertex u to v 1 , and all the edges v i v i+1 for i = 1, 2, . . . , n − 1, and the edge v 1 v n . Let η be the constant function η ≡ 1, let X = {v n } and let η ′ (v n ) = 0, which is an index function of G[X]. Then η ′ is a non-singular index function of G[X]. To prove that pind(A G ) = 1, i.e., η is a non-singular index function of G, it suffices, by Corollary 1, to show that for each vertex v, This is a routine check. Assume first that v is not a leaf of T .
Next, consider the case that v is a leaf of T .
A Halin graph is a planar graph obtained by taking a plane tree (an embedding of a tree on the plane) without degree 2 vertices by adding a cycle connecting the leaves of the tree cyclically. 2 η(v) = 1 for all vertices v, except that η(n, 1) = 0, and η((n, j)) = 2 for 2 ≤ j ≤ m.
Then η is non-singular for G.
Assume η(n, 1) = 0 and η(n, j) = 2 for 2 ≤ j ≤ m. We delete vertices (n, m), (n, m − 1), . . . , (n, 1) in this order, and need not to change η except for while deleting (n, 2), we increase η(n, 1) by 1. It follows from induction hypothesis that the resulting index function is non-singular for P n−1 ✷P m , and by Theorem 1 that the original index function η is nonsingular for G.