منابع مشابه
Internal Partitions of Regular Graphs
An internal partition of an n-vertex graph G = (V,E ) is a partition of V such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every d -regular graph with n > N(d ) vertices has an internal partition. Here we prove this for d = 6. The case d = n − 4 is of particular interest and leads to interesting new open problems on cubic g...
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A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...
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The path partition number of a graph is the minimum number of paths required to partition the vertices. We consider upper bounds on the path partition number under minimum and maximum degree assumptions.
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In this article we study the monochromatic cycle partition problem for non-complete graphs. We consider graphs with a given independence number (G)= . Generalizing a classical conjecture of Erd” os, Gyárfás and Pyber, we conjecture that if we r -color the edges of a graph G with (G)= , then the vertex set of G can be partitioned into at most r vertex disjoint monochromatic cycles. In the direct...
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A complete partition of a graph is a partition of the vertex set such that any two classes are connected by an edge. We consider the problem of finding a complete partition maximizing the number of classes. This relates to clustering into the greatest number of groups so as to minimize the diameter (inter-cluster connectivity) without concern for the intra-cluster topology. We give a randomized...
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ژورنال
عنوان ژورنال: Combinatorics, Probability and Computing
سال: 2020
ISSN: 0963-5483,1469-2163
DOI: 10.1017/s0963548320000553