Abstract:The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when {i, j} is an edge in G for i ≠ j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force all vertices of the graph blue by applying the color change rule. This research is motivated by the longstanding question of characterizing graphs G for which M(G) = Z(G). The following conjecture was proposed at the 2017 AIM workshop Zero forcing and its applications: If G is a bipartitesemiregular graph, then M(G) = Z(G). A counterexample was found by J. C.-H. Lin but questions remained as to which bipartite 3-semiregular graphs have M(G) = Z(G). We use various tools to nd bipartite families of graphs with regularity properties for which the maximum nullity is equal to the zero forcing number; most are bipartite 3-semiregular. In particular, we use the techniques of twinning and vertex sums to form new families of graphs for which M(G) = Z(G) and we additionally establish M(G) = Z(G) for certain Generalized Petersen graphs.
Given a graph G, an exact r-coloring of G is a surjective functionAn arithmetic progression is rainbow if all of the vertices are colored distinctly. The fewest number of colors that guarantees a rainbow arithmetic progression of length three is called the anti-van der Waerden number of G and is denoted aw(G, 3). It is known that 3 ≤ aw(G H, 3) ≤ 4. Here we determine exact values aw(T T ′ , 3) for some trees T and T ′ , determine aw(G T, 3) for some trees T , and determine aw(G H, 3) for some graphs G and H.
Let C be a family of edge-colored graphs. A t-edge colored graph G is (C, t)saturated if G does not contain any graph in C but the addition of any edge in any color in [t] creates a copy of some graph in C. Similarly to classical saturation functions, define sat t (n, C) to be the minimum number of edges in a (C, t) saturated graph. Let C r (H) be the family consisting of every edge-colored copy of H which uses exactly r colors.In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for sat t (n, C r (K k )) for all r, showing a sharp change in behavior when r ≥ k−1 2 + 2. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine sat t (n, C 2 (K 3 )) exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.
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