7,106 research outputs found
A General Stabilization Bound for Influence Propagation in Graphs
We study the stabilization time of a wide class of processes on graphs, in
which each node can only switch its state if it is motivated to do so by at
least a fraction of its neighbors, for some . Two examples of such processes are well-studied dynamically changing
colorings in graphs: in majority processes, nodes switch to the most frequent
color in their neighborhood, while in minority processes, nodes switch to the
least frequent color in their neighborhood. We describe a non-elementary
function , and we show that in the sequential model, the worst-case
stabilization time of these processes can completely be characterized by
. More precisely, we prove that for any ,
is an upper bound on the stabilization time of
any proportional majority/minority process, and we also show that there are
graph constructions where stabilization indeed takes
steps
Extremal \u3cem\u3eH\u3c/em\u3e-Colorings of Trees and 2-connected Graphs
For graphs G and H, an H-coloring of G is an adjacency preserving map from the vertices of G to the vertices of H. H-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of H-colorings. In this paper, we prove several results in this area. First, we find a class of graphs H with the property that for each H∈H, the n-vertex tree that minimizes the number of H -colorings is the path Pn. We then present a new proof of a theorem of Sidorenko, valid for large n, that for every H the star K1,n−1 is the n-vertex tree that maximizes the number of H-colorings. Our proof uses a stability technique which we also use to show that for any non-regular H (and certain regular H ) the complete bipartite graph K2,n−2 maximizes the number of H-colorings of n -vertex 2-connected graphs. Finally, we show that the cycle Cn has the most proper q-colorings among all n-vertex 2-connected graphs
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
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