Graphs are often used to model risk management in various systems. Particularly, Caskurlu et al. in [6] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. It is well-known that the vertex cover problem is in P on bipartite graphs; however, the computational complexity of the partial vertex cover problem on bipartite graphs is open. In this paper, we show that the partial vertex cover problem is NP-hard on bipartite graphs. Then, we show that the budgeted maximum coverage problem (a problem related to partial vertex cover problem) admits an 8 9approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation.

For bipartite graphs the NP-completeness is proved for the problem of existence of maximum matching which removal leads to a graph with given lower(upper)bound for the cardinality of its maximum matching.Comment: 12 pages, 8 figures. Discrete Mathematics, to appea

A normal k‐edge‐coloring of a cubic graph is a proper edge‐coloring with k colors having the additional property that when looking at the set of colors assigned to any edge e and the four edges adjacent to it, we have either exactly five distinct colors or exactly three distinct colors. We denote by χ N true′ ( G ) the smallest k, for which G admits a normal k‐edge‐coloring. Normal k‐edge‐colorings were introduced by Jaeger to study his well‐known Petersen Coloring Conjecture. More precisely, it is known that proving χ N true′ false( G false) ≤ 5 for every bridgeless cubic graph is equivalent to proving the Petersen Coloring Conjecture and then it implies, among others, Cycle Double Cover Conjecture and Berge‐Fulkerson Conjecture. Considering the larger class of all simple cubic graphs (not necessarily bridgeless), some interesting questions naturally arise. For instance, there exist simple cubic graphs, not bridgeless, with χ N true′ ( G ) = 7. In contrast, the known best general upper bound for χ N true′ ( G ) was 9. Here, we improve it by proving that χ N true′ ( G ) ≤ 7 for any simple cubic graph G, which is best possible. We obtain this result by proving the existence of specific nowhere zero Z 2 2‐flows in 4‐edge‐connected graphs.