Displaying similar documents to “Cores and shells of graphs”

On integral sum graphs with a saturated vertex

Zhibo Chen (2010)

Czechoslovak Mathematical Journal

Similarity:

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...

On well-covered graphs of odd girth 7 or greater

Bert Randerath, Preben Dahl Vestergaard (2002)

Discussiones Mathematicae Graph Theory

Similarity:

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth...

Join of two graphs admits a nowhere-zero 3 -flow

Saieed Akbari, Maryam Aliakbarpour, Naryam Ghanbari, Emisa Nategh, Hossein Shahmohamad (2014)

Czechoslovak Mathematical Journal

Similarity:

Let G be a graph, and λ the smallest integer for which G has a nowhere-zero λ -flow, i.e., an integer λ for which G admits a nowhere-zero λ -flow, but it does not admit a ( λ - 1 ) -flow. We denote the minimum flow number of G by Λ ( G ) . In this paper we show that if G and H are two arbitrary graphs and G has no isolated vertex, then Λ ( G ∨ H ) ≤ 3 except two cases: (i) One of the graphs G and H is K 2 and the other is 1 -regular. (ii) H = K 1 and G is a graph with at least one isolated vertex or a component whose every...

Paths with restricted degrees of their vertices in planar graphs

Stanislav Jendroľ (1999)

Czechoslovak Mathematical Journal

Similarity:

In this paper it is proved that every 3 -connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23 . Analogous results are stated for 3 -connected planar graphs of minimum degree 4 and 5 . Moreover, for every pair of integers n ≥ 3 , k ≥ 4 there is a 2 -connected planar graph such that every path on n vertices in it has a vertex of degree k .