An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f : E → {1, . . . , |E|} such that for any pair of adjacent vertices x and y, ff (e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ la (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, several sufficient conditions for χ la (H) ≤ χ la (G) are obtained, where H is obtained from G with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.
We define a dimension, called the finitely presented dimension, for modules and commutative rings. This dimension has nice properties when the ring in question is coherent. We then compare the finitely presented dimension with the global dimension and the weak global dimension.Introduction. Similar to the projective and flat dimensions, we define a dimension, called the finitely presented dimension, for modules and commutative rings. It measures how far away a module is from being finitely presented, and how far away a ring is from being Noetherian.In §1 we give the definitions and show some of the general properties. In §2, with the additional assumption of coherence, we show that the finitely presented dimension has the properties that we expect of a 'dimension'. We also point out the difference from the usual dimensions. In §3 we make a comparison of the global dimension, the weak global dimension, and the finitely presented dimension of a coherent ring, and get the relation
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