Zero forcing, tree width, and graph coloring

We show that certain types of zero-forcing sets for a graph give rise to chordal supergraphs and hence to proper colorings. Zero-forcing was originally defined to provide a bound for matrix minimum rank problems [1], but is interesting as a graph-theoretic notion in its own right [5], and has applications to mathematical physics, such as quantum systems [3]. There are different flavors of zero-forcing, many corresponding to a minimum rank graph parameter, and each is typically defined via assignments of the colors black and white to vertices and a color-change rule that allows changing white vertices to black [2]; the associated zero-forcing number is then the smallest cardinality among sets of vertices that when colored black originally allow the entire graph to become colored black via (repeated) application of the color-change rule (zero-forcing sets). Barioli et al. [2] showed that even treewidth can be defined as a zero-forcing parameter. Their proof uses a characterization of treewidth involving the game of cops and robbers. In this paper, we will show that a treewidth zero-forcing set Z for a graph G can be used to directly construct a |Z|-tree on the vertices of G that contains G as a subgraph. As an application, we will see that many different types of zero-forcing sets give easy constructions of proper colorings and proper list-colorings. For a given coloring of the vertices of a graph using black and white, the treewidth color-change rule was defined as follows (standard definitions are taken from Diestel’s Graph Theory [4]): Definition. Let B be the set consisting of all the black vertices. Let W1, . . . ,Wk be the sets of vertices of the k components of G−B. For each component i, 1 ≤ i ≤ k, let Ci ⊆ B be the subset of black vertices that are considered to be active with regard to that component, where initially each Ci = B. If w ∈ Wi and for each component X of G[Wi] − w there is a vertex uX ∈ Ci with no white neighbor in G[VX ∪ B], then change the color of w to black and associate to each connected component X of G[Wi]− w a new active set equal to (Ci − uX) ∪ {w}. L. MITCHELL /AUSTRALAS. J. COMBIN. 61 (1) (2015), 19–22 20 When this color-change rule is applied, we will say that the uX vertices force w and the uX vertices and w together comprise a forcing. In studying treewidth zero-forcing sets, we will find it advantageous to keep track of active sets for each vertex as well as the progress of the color-changes. If Z is a treewidth zero-forcing set of a graph G and m = |G − Z|, let w1, . . . , wm be the vertices of G − Z in the order in which they are turned black (there may be more than one such order – we’ll pick one). Our notational scheme will be subscripts that refer to the progress of the forcing: a subscript i, 1 ≤ i ≤ m, will reference the state of things after i forces, that is, when wi has become black and (if i < m) wi+1 is still white. For example, let B0 = Z and recursively define, for 1 ≤ i ≤ m, Bi = Bi−1 ∪ {wi}. Then Bi is the set of black vertices after i forces. Continuing in this spirit, if u is a vertex of G − Bj for some j with 0 ≤ j ≤ m − 1, let C j be the connected component of G−Bj containing u and let Aj be the set of active vertices of C j . Proposition. Let Z be a treewidth zero-forcing set of a graphG and use the notation above. Let G0 be the graph obtained from G by adding edges between any two vertices of B0 that are not adjacent in G. Let Gi be the graph obtained from Gi−1 by adding edges between wi and any vertices of A wi i−1 that are not neighbors of wi in Gi. Then Gm is a |Z|-tree on the same vertices as G containing G as a subgraph. Moreover, Z is a treewidth zero-forcing set for Gm with the same forcings in the same order. Proof. Since no vertices are added and no edges are removed, G is a subgraph of Gm and they share the same vertex set. To prove that Gm is a |Z|-tree, we will use the recursive definition of k-tree. Specifically, we claim that for each i with 1 ≤ i ≤ m, each Gi[Bi] is a |Z|-tree and that Gi[Bi] is obtained from Gi−1[Bi−1] by adding the vertex wi, which is adjacent in Gi[Bi] to the vertices of a |Z|-clique in Gi−1[Bi−1]. We begin by collecting some useful facts. First, active sets (for both vertices and components) start with |Z| vertices and only change via a one-for-one swap of vertices, so |Ai i−1| = |Z| for each i. Let N(w) denote the set of neighbors of vertex w in G. We next claim that N(wi) ∩ Bi−1 ⊆ Ai i−1 for each i with 1 ≤ i ≤ m. To see this, suppose that v ∈ N(wi) ∩ Bi−1. There are two possibilities: either v ∈ B0, in which case v ∈ Ai 0 , or v / ∈ B0, meaning v = wj for some j < i. In the latter case, since wi and v = wj are adjacent in G, Ci j−1 = C wj j−1, and so v = wj ∈ Ai j due to the jth force. Either way, v is at some point an active vertex for wi. Suppose v / ∈ Ai i−1. Then for some k < i (and k > j if v = wj), v was uX for X = C wi k , but this contradicts that v ∈ N(wi) since wi would be a white neighbor of v in G[VX ∪Bk−1]. Thus v ∈ Ai i−1. Finally, we claim that for i and j such that 0 ≤ i < j ≤ m, each A i forms a clique in each Gk such that 0 ≤ k < j. First, note that A0 consists of the vertices of Z, which form a clique in G0 by definition. Assume then that 0 < i < j ≤ m and the vertices of the set A wj i−1 form a clique in Gi−1. If A wj i−1 = A wj i , the vertices of A wj i will still be a clique in Gi. If A wj i−1 = A i , then wi ∈ C i−1. Thus Ci i−1 = C i−1, which L. MITCHELL /AUSTRALAS. J. COMBIN. 61 (1) (2015), 19–22 21 by the definition of the active sets for vertices implies Ai i−1 = A wj i−1. By assumption, Ai i−1 is a clique in Gi−1. By construction, A wi i−1 ∪ {wi} is a clique in Gi, and, by definition, A wj i ⊂ Ai i−1 ∪ {wi}, so that A i is also a clique in Gi. The claim follows by induction. To start the induction for the main part of the proof, notice that G0 is a |Z|-clique in G0[B0] = G0 by construction, that A w1 0 = B0 by definition, and thus w1 ∈ B1 is adjacent in G1[B1] to the vertices of G0 by construction. Thus G1[B1] is a |Z|-tree. Suppose now that Gj[Bj ] is a |Z|-tree for some j with 1 ≤ j < m. By construction, the vertices of Gj+1[Bj+1] are those of Gj [Bj] and the vertex wj+1, which by construction and the second fact above is adjacent to exactly the vertices of A wj+1 j in Gj+1[Bj+1]. By the first fact above, |A j | = |Z|, and by the third and final fact above, Gj[A wj+1 j ] is a clique. Thus Gj+1[Bj+1] is a |Z|-tree. By induction, Gm = Gm[Bm] is a |Z|-tree. We also claim that Z is a zero-forcing set of Gm using the same forces. The only way this can fail to be true is if an edge is added to G that will cause the color-change rule to no longer be applicable at some point. We will show this cannot happen. Consider a vertex z that is the uX vertex for a connected component X of G− Bi for some i such that 0 ≤ i < m. If an edge is added between z and a vertex that is in Bi, that edge does not affect the ability of z to be uX . Thus we only need consider an edge added between z and some wj where j > i. If C wj i = X , then the edge to wj does not affect the ability of z to be uX . If C wj i = X, then z is replaced by wi in A wj i . A vertex that has become inactive for another vertex can never become active again, so z / ∈ A i contradicts that z = uX , since z = uX implies z ∈ Ai i−1. In the hierarchy of color-change rules, the treewidth color-change rule is among the least restrictive. In particular, it is clear from the article by Barioli et al. [2] that, among others, standard zero-forcing sets and positive semidefinite zero-forcing sets are also treewidth zero-forcing sets. As an application of our proposition, many types of zero-forcing sets thus give proper colorings as in the following corollary: Corollary. If G is a graph with a treewidth zero-forcing set Z, where the vertices of G − Z are w1, . . . , wm in the order they are forced, given an assignment of a list of |Z|+ 1 colors to each vertex, a proper list-coloring of G may be selected by first choosing a proper list-coloring of G[Z] then selecting an available color from the list of each wi in order. Proof. A proper list-coloring of G[Z] exists since |G[Z]| < |Z|+1, and from the proof of the proposition, each wi will be adjacent to at most |Z| vertices whose colors have already been selected when its turn arrives. Remark. The treewidth color-change rule is significantly different from the original zero-forcing color-change rule in that its application requires knowledge of more than the graph and which vertices are black. We would be very interested to know if it possible to define a color-change rule that depends only on the graph and which vertices are black that will also give treewidth as its zero-forcing number. L. MITCHELL /AUSTRALAS. J. COMBIN. 61 (1) (2015), 19–22 22