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

1MS. S. Jai Roselin., Research Scholar, Department of Mathematics, St.Josephโ€™s college (Autonomous) affiliated to Bharathidasan University, Trichy.
2Dr. L. Benedict Michael Raj, Head and Associate Professor, St.Josephโ€™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: L34091081219/2019ยฉBEIESP | DOI: 10.35940/ijitee.L3409.1081219
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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:Given a graph ๐‘ฎ = (๐‘ฝ, ๐‘ฌ) and a finite set ๐‘ป 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. For a ๐‘ป-coloring ๐’‡ of G, the f-span ๐’”๐’‘๐‘ป ๐’‡ (๐‘ฎ) is the maximum value of |๐’‡(๐’–) โˆ’ ๐’‡(๐’˜)| over all pairs ๐’–, ๐’˜ of vertices of ๐‘ฎ. The ๐‘ป-span ๐’”๐’‘๐‘ป(๐‘ฎ) is the minimum ๐’‡-span over all ๐‘ป-colorings f of ๐‘ฎ. The ๐’‡-edge span ๐’†๐’”๐’‘๐‘ป ๐’‡ (๐‘ฎ) of a ๐‘ป-coloring is the maximum value of ๐’‡ ๐’– โˆ’ ๐’‡ ๐’˜ over all edges ๐’–๐’˜ of ๐‘ฎ. The ๐‘ป-edge span ๐’†๐’”๐’‘๐‘ป(๐‘ฎ) is the minimum ๐’‡-edge span over all ๐‘ป-colorings f of ๐‘ฎ. It is known that ๐’”๐’‘๐‘ป(๐‘ฏ) โ‰ค ๐’”๐’‘๐‘ป(๐‘ฎ) and ๐’†๐’”๐’‘๐‘ป(๐‘ฏ) โ‰ค e๐’”๐’‘๐‘ป(๐‘ฎ) for every graph ๐‘ฎ. In this paper we classify which graphs containing a sub graph ๐‘ฏ such that ๐’”๐’‘๐‘ป ๐‘ฏ < ๐’”๐’‘๐‘ป(๐‘ฎ) and ๐’†๐’”๐’‘๐‘ป(๐‘ฏ) < e๐’”๐’‘๐‘ป(๐‘ฎ). Also we discuss the Mycielskian of ๐‘ป-coloring. Keywords: T-coloring, T-span, T-edge SpanAMS Subject Classification 05C15
Scope of the Article: Classification