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A parity path in a vertex colouring of a graph is a path along which each colour is used an even number of times. Let χ p (G) be the least number of colours in a proper vertex colouring of G having no parity path. It is proved that for any graph G we have the following tight bounds χ(G) ≤ χ p (G) ≤ |V (G)| − α(G) + 1, where χ(G) and α(G) are the chromatic… (More)

We prove that for any two minor hereditary properties P 1 and P 2 , such that P 2 covers P 1 , and for any graph G ∈ P 2 there is a P 1-bipartition of G. Some remarks on minimal reducible bounds are also included.

The well-known lower bound on the independence number of a graph due 1981) can be established as a performance guarantee of two natural and simple greedy algorithms or of a simple randomized algorithm. We study possible generalizations and improvements of these approaches using vertex weights and discuss conditions on so-called potential functions p G : V… (More)

In this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function $${\omega }$$ ω provides for each vertex $$v$$ v the minimum number of searchers required to guard $$v$$ v , i.e., the fugitive may not pass through $$v$$ v without being detected only if at least $${\omega }(v)$$… (More)

A complete k-coloring of a graph G = (V, E) is an assignment ϕ : V → {1,. .. , k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in… (More)