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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, f+(x) ≠ f+(y), where the induced vertex label f+(x) = ∑f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. Let χ(G) be the chromatic number of G. In this paper, sharp upper and lower bounds of χla(G) for G with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for k ≥ 1, there are infinitely many graphs with k ≥ χ(G) - 1 pendant vertices and χla(G) = k + 1. The work of this paper leads to conjecture and problem on graphs with pendant(s). [ABSTRACT FROM AUTHOR]

A proper edge k-colouring of a graph $$G=(V,E)$$ is an assignment $$c:E\rightarrow \{1,2,\ldots ,k\}$$ of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge k-colouring, or nsd k-colouring for short, is a proper edge k-colouring such that $$\sum _{e\ni u}c(e)\ne \sum _{e\ni v}c(e)$$ for every edge uv of G. We denote by $$\chi '_{\Sigma }(G)$$ the neighbour sum distinguishing index of G, which is the least integer k such that an nsd k-colouring of G exists. By definition at least maximum degree, $$\Delta (G)$$ colours are needed for this goal. In this paper we prove that $$\chi '_\Sigma (G) \le \Delta (G)+1$$ for any graph G without isolated edges, with $$\mathrm{mad}(G)<3$$ and $$\Delta (G) \ge 6$$ . [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new class of graceful graphs. In particular, we show that linear cyclic snakes (1,k)C4- snake and (2,k)C4- snake are graceful. We also prove that linear cyclic snakes (1,k)C8- snake and (2,k)C8- snake are graceful. We generalize the above results “ the linear cyclic snakes (m, k)C4- snake and (m, k)C8- snake are graceful “. Finally, we introduce a new conjecture” All the linear cyclic snakes (m, k)Cn-snakes are grace- ful ifn≡ 0 (mod 4 ) orn≡ 3 (mod 4 )”. [ABSTRACT FROM AUTHOR]