A graph G = (V, E) is antimagic if there is a one-to-one correspondence f : E → {1, 2, . . . , |E|} such that for any two vertices u, v,It is known that bipartite regular graphs are antimagic and nonbipartite regular graphs of odd degree at least three are antimagic. Whether all non-bipartite regular graphs of even degree are antimagic remained an open problem. In this paper, we solve this problem and prove that all even degree regular graphs are antimagic.

Suppose r ! 2 is a real number. A proper r-flow of a directed multi-graphG G ¼ ðV ; EÞ is a mapping f : E ! R such that (i) for every edge e 2 E, 1 jf ðeÞj r À 1; (ii) for every vertex v 2 V , P e2E þðvÞ f ðeÞÀ P e2E ÀðvÞ f ðeÞ ¼ 0. The circular flow number of a graph G is the least r for which an orientation of G admits a proper r-flow. The well-known 5-flow conjecture is equivalent to the statement that every bridgeless graph has circular flow number at most 5. In this paper, we prove that for any rational number r between 2 and 5, there exists a graph G with circular flow number r. ß

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