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In a zero forcing process , an initial vertex coloring of a graph is updated iteratively according to the following conversion rule: an uncolored vertex becomes colored if it is the only uncolored neighbor of some colored vertex. In this process, a zero forcing set is an initial set of colored vertices such that all the remaining vertices will ultimately change from uncolored to colored, and the zero forcing number is the minimum cardinality of a zero forcing set. We investigate two related concepts: a zero blocking set is the complement of a set which is not a zero forcing set, and the zero blocking number is the minimum cardinality of a zero blocking set. We provide a tight upper bound for the zero blocking number of Cartesian products of two arbitrary graphs. Exact values that are smaller than this general upper bound are also established when one of the graphs in the product is a cycle and the other is a path with at least as many vertices as the cycle. [ABSTRACT FROM AUTHOR]

Given a positive integer d, an L(d, 1)-labeling of a graph G is an assignment of nonnegative integers to its vertices such that adjacent vertices must receive integers at least d apart, and vertices at distance two must receive integers at least one apart. The λd-number of G is the minimum k so that G has an L(d, 1)-labeling using labels in {0, 1, ..., k}. Informally, an amalgamation of two disjoint graphs G1 and G2 along a fixed graph G0 is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to G0, one in G1 and the other in G2. A flower is an amalgamation of two or more cycles along a single vertex. We provide the exact λ2-number of a generalized flower which is the Cartesian product of a path Pn and a flower, or equivalently, an amalgamation of cylindrical rectangular grids along a certain Pn. In the process, we provide general upper bounds for the λd-number of the Cartesian product of P n and any graph G, using circular L(d+1, 1)-labelings of G where the labels {0, 1, ..., k} are arranged sequentially in a circle and the distance between two labels is the shortest distance on the circle. [ABSTRACT FROM AUTHOR]

The spectrum allocation problem in wireless communications can be modeled through vertex labelings of a graph, wherein each vertex represents a transmitter and edges connect vertices whose corresponding transmitters are operating in close proximity. One well-known model is the L ( 2 , 1 ) -labeling of a graph G in which a function f maps the vertices of G to the nonnegative integers such that if vertices x and y are adjacent, then | f ( x ) − f ( y ) | ≥ 2 , and if x and y are at distance two, then | f ( x ) − f ( y ) | ≥ 1 . The λ -number of G is the minimum span over all L ( 2 , 1 ) -labelings of G . Informally, an amalgamation of two graphs G 1 and G 2 along a fixed graph G 0 is the simple graph obtained by identifying the vertices of two induced subgraphs isomorphic to G 0 , one in G 1 and the other in G 2 . In this work, we supply a tight upper bound for the λ -number of amalgamations of several Cartesian products of complete graphs along a complete graph and find the exact λ -numbers for certain infinite subclasses of amalgamations of this form. A surprising relationship between the former upper bound and the minimum makespan scheduling problem is highlighted. [ABSTRACT FROM AUTHOR]

In a zero forcing process , an initial vertex coloring of a graph is updated iteratively according to the following conversion rule: an uncolored vertex becomes colored if it is the only uncolored neighbor of some colored vertex. In this process, a zero forcing set is an initial set of colored vertices such that all the remaining vertices will ultimately change from uncolored to colored, and the zero forcing number is the minimum cardinality of a zero forcing set. We investigate two related concepts: a zero blocking set is the complement of a set which is not a zero forcing set, and the zero blocking number is the minimum cardinality of a zero blocking set. We provide upper and lower bounds for the zero blocking number of rectangular grids and discuss conditions under which these bounds coincide. We go on to use the same machinery to provide similar results for certain cylindrical and Möbius grids. [ABSTRACT FROM AUTHOR]

For a positive integer , a radio k-labeling of a graph is a function from its vertex set to the non-negative integers such that for all pairs of distinct vertices and , we have where is the distance between the vertices and in . The minimum span over all radio -labelings of is called the radio k-chromatic number and denoted by . The most extensively studied cases are the classic vertex colorings (), (2,1)-labelings (), radio labelings (, the diameter of ), and radio antipodal labelings (. Determining exact values or tight bounds for is often non-trivial even within simple families of graphs. We provide general lower bounds for for all cycles when and show that these bounds are exact values when . [ABSTRACT FROM AUTHOR]