Displaying similar documents to “ T -preserving homomorphisms of oriented graphs”

Set colorings in perfect graphs

Ralucca Gera, Futaba Okamoto, Craig Rasmussen, Ping Zhang (2011)

Mathematica Bohemica

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For a nontrivial connected graph G , let c : V ( G ) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v ∈ V ( G ) , the neighborhood color set NC ( v ) is the set of colors of the neighbors of v . The coloring c is called a set coloring if NC ( u ) ≠ NC ( v ) for every pair u , v of adjacent vertices of G . The minimum number of colors required of such a coloring is called the set chromatic number χ s ( G ) . We show that the decision variant of determining χ s ( G ) is NP-complete in the general case, and show that...

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

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As introduced by F. Harary in 1994, a graph G is said to be an i n t e g r a l s u m g r a p h if its vertices can be given a labeling f with distinct integers so that for any two distinct vertices u and v of G , u v is an edge of G if and only if f ( u ) + f ( v ) = f ( w ) for some vertex w in G . We prove that every integral sum graph with a saturated vertex, except the complete graph K 3 , has edge-chromatic number equal to its maximum degree. (A vertex of a graph G is said to be if it is adjacent to every...

Set vertex colorings and joins of graphs

Futaba Okamoto, Craig W. Rasmussen, Ping Zhang (2009)

Czechoslovak Mathematical Journal

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For a nontrivial connected graph G , let c V ( G ) → ℕ be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G , the neighborhood color set NC ( v ) is the set of colors of the neighbors of v . The coloring c is called a set coloring if NC ( u ) ≠ NC ( v ) for every pair u , v of adjacent vertices of G . The minimum number of colors required of such a coloring is called the set chromatic number χ s ( G ) . A study is made of the set chromatic number of the join G + H of two graphs G and H . Sharp lower...

Cores and shells of graphs

Allan Bickle (2013)

Mathematica Bohemica

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The k -core of a graph G , C k ( G ) , is the maximal induced subgraph H ⊆ G such that δ ( G ) ≥ k , if it exists. For k > 0 , the k -shell of a graph G is the subgraph of G induced by the edges contained in the k -core and not contained in the ( k + 1 ) -core. The core number of a vertex is the largest value for k such that v ∈ C k ( G ) , and the maximum core number of a graph, C ^ ( G ) , is the maximum of the core numbers of the vertices of G . A graph G is k -monocore if C ^ ( G ) = δ ( G ) = k . This paper discusses some basic results on the structure of k -cores and...

On multiset colorings of graphs

Futaba Okamoto, Ebrahim Salehi, Ping Zhang (2010)

Discussiones Mathematicae Graph Theory

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A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χₘ(G) of G. For every graph G, χₘ(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that for every pair k, r of integers with 2 ≤ k ≤ r - 1, there exists an r-regular graph with multiset...