L34091081219

T-Span, T-Edge Span Critical Graphs
Jai Roselin.S¹, Benedict Michael Raj.L^2
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