Clique numbers of graphs and irreducible exact m-covers of the integers
نویسندگان
چکیده
منابع مشابه
CLIQUE NUMBERS OF GRAPHS AND IRREDUCIBLE EXACT m-COVERS OF Z
For each m ≥ 1, we construct a graph G = (V, E) with ω(G) = m such that max 1≤i≤k ω(G[Vi]) = m for arbitrary partition V = V1 ∪ · · · ∪ Vk, where ω(G) is the clique number of G and G[Vi] is the induced graph of Vi. Using this result, we show that for each m ≥ 2 there exists an exact m-cover of Z which is not the union of two 1-covers.
متن کاملIrreducible Exact Covering Systems and Clique Numbers of Graphs
For each m ≥ 1, we construct a graph G = (V, E) with ω(G) = m such that max i=1,2 ω(ind(Vi)) = m for arbitrary partition V = V1 ∪ V2, where ω(G) is the clique number of G and ind(Vi) is the induced graph of Vi. Using this result, we may show that for each m ≥ 2 there exists an exact m-cover of Z which is not the union of two 1-covers.
متن کاملRegular clique covers of graphs
A family of cliques in a graph G is said to be p-regular if any two cliques in the family intersect in exactly p vertices. A graph G is said to have a p-regular k-clique cover if there is a p-regular family H of k-cliques of G such that each edge of G belongs to a clique in H. Such a p-regular kclique cover is separable if the complete subgraphs of order p that arise as intersections of pairs o...
متن کاملClique numbers of graphs
Many graph-theoretical problems involve the study of cliques, i .e ., complete subgraphs (not necessarily maximal) . In this context the following combinatorial problem arises naturally: For which numbers n and c is there a graph with n vertices and exactly c cliques? For fixed n, let G(n) denote the set of all such `clique numbers' c . Since each singleton and the empty set are always cliques,...
متن کاملchromatic and clique numbers of a class of perfect graphs
let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 2009
ISSN: 0196-8858
DOI: 10.1016/j.aam.2008.09.004