Odd induced subgraphs in graphs of maximum degree three

A long-standing conjecture asserts the existence of a positive constant c such that every simple graph of order n without isolated vertices contains an induced subgraph of order at least cn such that all degrees in this induced subgraph are odd. Radcliffe and Scott have proved the conjecture for trees, essentially with the constant c = 2/3. Scott proved a bound for c depending on the chromatic number. For general graphs it is known only that c, if it exists, is at most 2/7. In this paper, we prove that for graphs of maximum degree three, the theorem is true with c = 2/5, and that this bound is best possible. Gallai proved that in any graph there is a partition of the vertices into two sets so that the sub graph induced by each set has each vertex of even degree; also there is a partition so that one induced subgraph has all degrees even and the other has degrees odd. (See [3] problem 17.) Clearly we can not assure a partition in which each subgraph has all degrees odd. The weaker question then arises whether every simple graph contains a "large" induced subgraph with all degrees odd. We say that an odd subgraph of G is an induced subgraph H such that every vertex of H has odd degree in H. We use f(G) to denote the maximum order of an odd subgraph of G. (To avoid trivial cases, we will restrict G to be without isolated vertices.) We may thus state the conjecture in the form that there exists a positive constant c such that for an n-vertex graph G, f( G) :2: cn. (This conjecture is cited by Caro [2] as "part of the graph theory folklore" .) Caro [2] proved a weaker conjecture of Alon that for an n-vertex graph G, f (G) :2: cvn. Scott [5) improved this, proving that f(G) :2: cn/log(n). Radcliffe and Scott [4] have proved the original conjecture for trees, essentially with the constant c = 2/3. In general it is known [2) only that c, if it exists, is at most 2/7. In [5) Scott proves a bound for c based on the chromatic number of G. It follows immediately from this bound that for a graph of maximum degree three f( G) :2: n/3. Australasian Journal of Combinatorics 15(1997), pp.81-85 In this paper, we prove the best possible bound for graphs of maximum degree