Strong T-Coloring of Graphs
S. Jai Roselin1, L. Benedict Michael Raj2, K.A. Germina3
1S. Jai Roselin, Research scholar, Department of Mathematics, St.Josephโ€™s college (Autonomous) affiliated to Bharathidasan University, Trichy.
2L. Benedict Michael Raj , Associate Professor, St.Josephโ€™s college (Autonomous) affiliated to Bharathidasan University, Trichy.
3K.A. Germina, Associate Professor, Central University of Kerala, India.
Manuscript received on September 16, 2019. | Revised Manuscript received on 24 September, 2019. | Manuscript published on October 10, 2019. | PP: 4677-4681 | Volume-8 Issue-12, October 2019. | Retrieval Number: L3575081219/2019ยฉBEIESP | DOI: 10.35940/ijitee.L3575.1081219
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ยฉ The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: A ๐‘ป-coloring of a graph ๐‘ฎ = (๐‘ฝ,๐‘ฌ) is the generalized coloring of a graph. Given a graph ๐‘ฎ = (๐‘ฝ, ๐‘ฌ) and a finite set T of positive integers containing ๐ŸŽ , a ๐‘ป-coloring of ๐‘ฎ is a function ๐’‡ โˆถ ๐‘ฝ (๐‘ฎ) โ†’ ๐’ + โˆช {๐ŸŽ} for all ๐’– โ‰  ๐’˜ in ๐‘ฝ (๐‘ฎ) such that if ๐’–๐’˜ โˆˆ ๐‘ฌ(๐‘ฎ) then |๐’‡(๐’–) โˆ’ ๐’‡(๐’˜)| โˆ‰ ๐‘ป. We define Strong ๐‘ป-coloring (S๐‘ป-coloring , in short), as a generalization of ๐‘ป-coloring as follows. Given a graph ๐‘ฎ = (๐‘ฝ, ๐‘ฌ) and a finite set ๐‘ป of positive integers containing ๐ŸŽ, a S๐‘ป-coloring of ๐‘ฎ is a function ๐’‡ โˆถ ๐‘ฝ (๐‘ฎ) โ†’ ๐’ + โˆช {๐ŸŽ} for all ๐’– โ‰  ๐’˜ in ๐‘ฝ (๐‘ฎ) such that if ๐’–๐’˜ โˆˆ ๐‘ฌ(๐‘ฎ) then |๐’‡(๐’–) โˆ’ ๐’‡(๐’˜)| โˆ‰ ๐‘ป and |๐’‡(๐’–) โˆ’ ๐’‡(๐’˜)| โ‰  |๐’‡(๐’™) โˆ’ ๐’‡(๐’š)| for any two distinct edges ๐’–๐’˜, ๐’™๐’š in ๐‘ฌ(๐‘ฎ). The S๐‘ป-Chromatic number of ๐‘ฎ is the minimum number of colors needed for a S๐‘ป-coloring of ๐‘ฎ and it is denoted by ๐Œ๐‘บ๐‘ป(๐‘ฎ) . For a S๐‘ป coloring ๐’„ of a graph ๐‘ฎ we define the ๐’„๐‘บ๐‘ป- span ๐’”๐’‘๐‘บ๐‘ป ๐’„ (๐‘ฎ) is the maximum value of |๐’„(๐’–) โˆ’ ๐’„(๐’—)| over all pairs ๐’–, ๐’— of vertices of ๐‘ฎ and the S๐‘ป -span ๐’”๐’‘๐‘บ๐‘ป(๐‘ฎ) is defined by ๐’”๐’‘๐‘บ๐‘ป(๐‘ฎ) = min ๐’”๐’‘๐‘บ๐‘ป ๐’„ (๐‘ฎ) where the minimum is taken over all ST-coloring c of G. Similarly the ๐’„๐‘บ๐‘ป-edgespan ๐’†๐’”๐’‘๐‘บ๐‘ป ๐’„ (๐‘ฎ) is the maximum value of |๐’„(๐’–) โˆ’ ๐’„(๐’—)| over all edges ๐’–๐’— of ๐‘ฎ and the S๐‘ป-edge span ๐’†๐’”๐’‘๐‘บ๐‘ป(๐‘ฎ) is defined by ๐’†๐’”๐’‘๐‘บ๐‘ป(๐‘ฎ) = min ๐’†๐’”๐’‘๐‘บ๐‘ป ๐’„ ๐‘ฎ where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S๐‘ป- chromatic number, ๐’”๐’‘๐‘บ๐‘ป(๐‘ฎ) , and ๐’†๐’”๐’‘๐‘บ๐‘ป(๐‘ฎ) of graphs.
Keywords: ๐‘ป-coloring, S๐‘ป-coloring, Span, Edge Span. AMS Subject Classification 05C15
Scope of the Article: Classification

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