On dynamic monopolies of graphs with general thresholds

Let $G$ be a graph and ${\mathcal{\tau}}: V(G)\rightarrow \Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is said to be dynamic monopoly (or simply dynamo) if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i=1,..., k-1$ each vertex $v$ in $D_{i+1}$ has at least $t(v)$ neighbors in $D_0\cup ...\cup D_i$. Dynamic monopolies are in fact modeling the irreversible spread of influence such as disease or belief in social networks. We denote the smallest size of any dynamic monopoly of $G$, with a given threshold assignment, by $dyn(G)$. In this paper we first define the concept of a resistant subgraph and show its relationship with dynamic monopolies. Then we obtain some lower and upper bounds for the smallest size of dynamic monopolies in graphs with different types of thresholds. Next we introduce dynamo-unbounded families of graphs and prove some related results. We also define the concept of a homogenious society that is a graph with probabilistic thresholds satisfying some conditions and obtain a bound for the smallest size of its dynamos. Finally we consider dynamic monopoly of line graphs and obtain some bounds for their sizes and determine the exact values in some special cases

First-Fit coloring of Cartesian product graphs and its defining sets

Let the vertices of a Cartesian product graph $G\Box H$ be ordered by an ordering $\sigma$. By the First-Fit coloring of $(G\Box H, \sigma)$ we mean the vertex coloring procedure which scans the vertices according to the ordering $\sigma$ and for each vertex assigns the smallest available color. Let $FF(G\Box H,\sigma)$ be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether $FF(G\Box H,\sigma)=FF(G\Box H,\tau)$, where $\sigma$ and $\tau$ are arbitrary orders. We study and obtain some bounds for $FF(G\Box H,\sigma)$, where $\sigma$ is any quasi-lexicographic ordering. The First-Fit coloring of $(G\Box H, \sigma)$ does not always yield an optimum coloring. A greedy defining set of $(G\Box H, \sigma)$ is a subset $S$ of vertices in the graph together with a suitable pre-coloring of $S$ such that by fixing the colors of $S$ the First-Fit coloring of $(G\Box H, \sigma)$ yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of $G\Box H$ with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in $G\Box H$, including some extremal results for Latin squares.Comment: Accepted for publication in Contributions to Discrete Mathematic

Bounds for chromatic number in terms of even-girth and booksize

AbstractThe even-girth of any graph G is the smallest length of any even cycle in G. For any two integers t,k with 0≤t≤k−2, we denote the maximum number of cycles of length k such that each pair of cycles intersect in exactly a unique path of length t by bt,k(G). This parameter is called the (t,k)-booksize of G. In this paper we obtain some upper bounds for the chromatic and coloring numbers of graphs in terms of even-girth and booksize. We also prove some bounds for graphs which contain no cycle of length t where t is a small and fixed even integer