We answer in the negative a question of Oporowski and Zhao [Discrete Math., 309 (2009), pp. 2948–2951] asking whether every graph with crossing number at most 5 and clique number at most 5 is 5-colorable. However, we show that every graph with crossing number at most 4 and clique number at most 5 is 5-colorable. We also show some colorability results on graphs that can be made planar by removing a few edges. In particular, we show that, if a graph with clique number at most 5 has three edges whose removal leaves the graph planar, then it is 5-colorable.

For a given number L , an L -length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s - t -path of length at most L remains in the graph after removing the edges (nodes, respectively) in C . An L -length-bounded flow is a flow that can be decomposed into flow paths of length at most L . In contrast to classical flow theory, we describe instances for which the minimum L -length-bounded edge-cut (node-cut, respectively) is Θ( n 2/3 )-times (Θ(√ n )-times, respectively) larger than the maximum L -length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is NP -hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of node-cuts and for L ≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O (min{ L , n/L }) ⊆ O √ n in the node case and O (min { L , n 2 / L 2 ,√ m } ⊆ O 2/3 in the edge case, where m denotes the number of edges. Concerning L -length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is NP -complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.