A Note on Bipartite Graphs without 2k-Cycles
نویسندگان
چکیده
We address the question of the maximum number ex(m,n, C2k) of edges in an m by n bipartite graph without a cycle of length 2k. We prove that for each k ≥ 2, ex(m,n, C2k) ≤ (2k − 3) [ (mn) k+1 k + m + n ] if k is odd. (2k − 3) [ m k+2 k n 1 2 + m + n ] if k is even.
منابع مشابه
Properties of Certain Families of 2k-Cycle-Free Graphs
Let v = v(G) and e = e(G) denote the order and size of a simple graph G, respectively. Let G = {Gi}i≥1 be a family of simple graphs of magnitude r > 1 and constant λ > 0, i.e. e(Gi) = (λ+ o(1))v(Gi), i→∞. For any such family G whose members are bipartite and of girth at least 2k + 2, and every integer t, 2 ≤ t ≤ k − 1, we construct a family G̃t of graphs of same magnitude r, of constant greater ...
متن کاملA result on 2k-cycle-free bipartite graphs
Let G be a bipartite graph with vertex parts of orders N and M , and X edges. I prove that if G has no cycles of length 2l, for all l ∈ [2, 2k], and N ≥ M , then X < M 1 2 N k+1 2k + O(N).
متن کاملA Note on Graphs Without Short Even Cycles
In this note, we show that any n-vertex graph without even cycles of length at most 2k has at most 12n 1+1/k + O(n) edges, and polarity graphs of generalized polygons show that this is asymptotically tight when k ∈ {2, 3, 5}.
متن کاملA minimum degree result for disjoint cycles and forests in bipartite graphs
Let F = (U1, U2;W ) be a forest with |U1| = |U2| = s, where s ≥ 2, and let G = (V1, V2, E) be a bipartite graph with |V1| = |V2| = n ≥ 2k + s, where k is a nonnegative integer. Suppose that the minimum degree of G is at least k+ s. We show that if n > 2k+ s then G contains the disjoint union of the forest F and k disjoint cycles. Moreover, if n = 2k+ s, then G contains the disjoint union of the...
متن کاملDisjoint hamiltonian cycles in bipartite graphs
Let G = (X, Y ) be a bipartite graph and define σ 2(G) = min{d(x) + d(y) : xy / ∈ E(G), x ∈ X, y ∈ Y }. Moon and Moser [5] showed that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n + 1 then G is hamiltonian, sharpening a classical result of Ore [6] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n+ 2k− 1 then G contains k edge...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003