a b s t r a c tWe prove that every connected graph G of order n has a spanning tree T such that for every edge e of T the edge cut defined in G by the vertex sets of the two components of T − e contains at most n 3 2 edges. This result solves a problem posed by Ostrovskii (M.I. Ostrovskii, Minimal congestion trees, Discrete Math. 285 (2004) 219-226).
We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most 3 due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, Discrete Math. 233 (2001), 233-237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n − m − 1)/7.
Abstract. We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g (g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g = 3, Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g ∈ {4, 5}, we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g ≥ 6, both problems are APX-hard already for graphs with maximum degree 3.
We prove that every Hamiltonian graph with n vertices and m edges has cycles with more thanFor general m and n, there exist such graphs having at most 2 √ p + 1 different cycle lengths.
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