Strong T-Coloring of Graphs
S. Jai Roselin1, L. Benedict Michael Raj2, K.A. Germina3
1S. Jai Roselin, Research scholar, Department of Mathematics, St.Joseph’s college (Autonomous) affiliated to Bharathidasan University, Trichy.
2L. Benedict Michael Raj , Associate Professor, St.Joseph’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: L3575081219/2019©BEIESP | DOI: 10.35940/ijitee.L3575.1081219
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© 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 𝑻-coloring of a graph 𝑮 = (𝑽,𝑬) is the generalized coloring of a graph. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set T of positive integers containing 𝟎 , a 𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻. We define Strong 𝑻-coloring (S𝑻-coloring , in short), as a generalization of 𝑻-coloring as follows. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set 𝑻 of positive integers containing 𝟎, a S𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻 and |𝒇(𝒖) − 𝒇(𝒘)| ≠ |𝒇(𝒙) − 𝒇(𝒚)| for any two distinct edges 𝒖𝒘, 𝒙𝒚 in 𝑬(𝑮). The S𝑻-Chromatic number of 𝑮 is the minimum number of colors needed for a S𝑻-coloring of 𝑮 and it is denoted by 𝝌𝑺𝑻(𝑮) . For a S𝑻 coloring 𝒄 of a graph 𝑮 we define the 𝒄𝑺𝑻- span 𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all pairs 𝒖, 𝒗 of vertices of 𝑮 and the S𝑻 -span 𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒔𝒑𝑺𝑻(𝑮) = min 𝒔𝒑𝑺𝑻 𝒄 (𝑮) where the minimum is taken over all ST-coloring c of G. Similarly the 𝒄𝑺𝑻-edgespan 𝒆𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all edges 𝒖𝒗 of 𝑮 and the S𝑻-edge span 𝒆𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒆𝒔𝒑𝑺𝑻(𝑮) = min 𝒆𝒔𝒑𝑺𝑻 𝒄 𝑮 where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S𝑻- chromatic number, 𝒔𝒑𝑺𝑻(𝑮) , and 𝒆𝒔𝒑𝑺𝑻(𝑮) of graphs.
Keywords: 𝑻-coloring, S𝑻-coloring, Span, Edge Span. AMS Subject Classification 05C15
Scope of the Article: Classification