A vertex coloring c : V β {1, 2, . . . , t} of a graph G = (V , E) is a vertex t-ranking if for any two vertices of the same color every path between them contains a vertex of larger color. The vertex ranking number Ο r (G) is the smallest value of t such that G has a vertex t-ranking. A Ο r (G)-ran
Vertex-Compressed Subalgebras of a Graph von
β Scribed by Ilwoo Cho
- Publisher
- Springer Netherlands
- Year
- 2008
- Tongue
- English
- Weight
- 741 KB
- Volume
- 108
- Category
- Article
- ISSN
- 0167-8019
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π SIMILAR VOLUMES
Let r(G) denote the rank of the adjacency matrix of a graph G. When a vertex and its incident edges are deleted from G, the rank of the resultant graph cannot exceed r(G) and can decrease by at most 2. The problem of determining the exact effect of adding a single vertex to a graph is more difficult
## Abstract We prove in this note that the linear vertexβarboricity of any planar graph is at most three, which confirms a conjecture due to Broere and Mynhardt, and others.
Lawrence [2, Theorem 3] and Borodin and Kostochka [1, Lemma 2' 1 both give the same theorem about vertex colorings of graphs (Corollary 1 below). But Lawrence's proof, although powerful, is a little long, and Borodin and Kostoehka state the result without a proof.
In this work, we prove that ΞΊ 2 (S n ) = 6(n -3) for n β₯ 4, where S n is the n-dimensional star graph.