# Certain Results on Prime and Prime Distance Labeling of Graphs

01 May 2020-Vol. 1531, Iss: 1, pp 012062

About: The article was published on 2020-05-01 and is currently open access. It has received None citation(s) till now. The article focuses on the topic(s): Prime (order theory).

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TL;DR: It is proved that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary, and numerous results regarding periodic colourings are obtained.

Abstract: Four colours are necessary and sufficient to colour all the integers so that any two with difference equal to a prime have different colours. We investigate the corresponding problem when the setD of prescribed differences is a proper subset of the primes. In particular, we prove that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary. Numerous results regarding periodic colourings are also obtained. However, the problem of characterizing those setsD which necessitate four colours remains open.

61 citations

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TL;DR: It is proved that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs arePrime distance graphs if and only if the Twin Prime Conjecture and dePolignac’s Conjectures are true, respectively.

Abstract: A graph G is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the dierence of their labels is prime (either 2 or odd). We prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Twin Prime Conjecture and dePolignac’s Conjecture are true, respectively. We give a characterization of 2-odd graphs in terms of edge colorings, and we use this characterization to determine which circulant graphs of the form Circ(n;f1;kg) are 2-odd and to prove results on circulant prime distance graphs.

5 citations

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Abstract: Were considered transcendental equations with trigonometric and hyperbolic functions. Were obtained two-sided estimates for all their roots. AMS Subject Classification: 65H99

1 citations

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19 Dec 2016TL;DR: This paper derives certain general results concerning prime distance labeling of some cycle related graphs in the context of some graph operations, namely, power, fusion, duplication and vertex switching in cycle \(C_n\).

Abstract: A graph G is a prime distance graph if its vertices can be labeled with distinct integers in such a way that for any two adjacent vertices, the absolute difference of their labels is a prime number. It is known that cycles and bipartite graphs are prime distance graphs. In this paper we derive certain general results concerning prime distance labeling. We also investigate prime distance labeling of some cycle related graphs in the context of some graph operations, namely, power, fusion, duplication and vertex switching in cycle \(C_n\).

1 citations

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