T-Span, T-Edge Span Critical Graphs
Jai Roselin.S1, Benedict Michael Raj.L2

1MS. S. Jai Roselin., Research Scholar, Department of Mathematics, St.Joseph’s college (Autonomous) affiliated to Bharathidasan University, Trichy.
2Dr. L. Benedict Michael Raj, Head and Associate Professor, St.Joseph’s college (Autonomous) affiliated to Bharathidasan University, Trichy

Manuscript received on September 16, 2019. | Revised Manuscript received on 24 September, 2019. | Manuscript published on October 10, 2019. | PP: 3898-3901 | Volume-8 Issue-12, October 2019. | Retrieval Number: L34091081219/2019©BEIESP | DOI: 10.35940/ijitee.L3409.1081219
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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: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. For a 𝑻-coloring 𝒇 of G, the f-span 𝒔𝒑𝑻 𝒇 (𝑮) is the maximum value of |𝒇(𝒖) − 𝒇(𝒘)| over all pairs 𝒖, 𝒘 of vertices of 𝑮. The 𝑻-span 𝒔𝒑𝑻(𝑮) is the minimum 𝒇-span over all 𝑻-colorings f of 𝑮. The 𝒇-edge span 𝒆𝒔𝒑𝑻 𝒇 (𝑮) of a 𝑻-coloring is the maximum value of 𝒇 𝒖 − 𝒇 𝒘 over all edges 𝒖𝒘 of 𝑮. The 𝑻-edge span 𝒆𝒔𝒑𝑻(𝑮) is the minimum 𝒇-edge span over all 𝑻-colorings f of 𝑮. It is known that 𝒔𝒑𝑻(𝑯) ≤ 𝒔𝒑𝑻(𝑮) and 𝒆𝒔𝒑𝑻(𝑯) ≤ e𝒔𝒑𝑻(𝑮) for every graph 𝑮. In this paper we classify which graphs containing a sub graph 𝑯 such that 𝒔𝒑𝑻 𝑯 < 𝒔𝒑𝑻(𝑮) and 𝒆𝒔𝒑𝑻(𝑯) < e𝒔𝒑𝑻(𝑮). Also we discuss the Mycielskian of 𝑻-coloring. Keywords: T-coloring, T-span, T-edge SpanAMS Subject Classification 05C15
Scope of the Article: Classification