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On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
A Study on Edge-Set Graphs of Certain Graphs
Let be a simple connected graph, with In this
paper, we define an edge-set graph constructed from the graph
such that any vertex of corresponds to the -th
-element subset of and any two vertices of
are adjacent if and only if there is at least one edge in the
edge-subset corresponding to which is adjacent to at least one edge
in the edge-subset corresponding to where are positive
integers. It can be noted that the edge-set graph of a graph
id dependent on both the structure of as well as the number of edges
We also discuss the characteristics and properties of the edge-set
graphs corresponding to certain standard graphs.Comment: 10 pages, 2 figure
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
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