T-Span, T-Edge Span Critical Graphs
Jai Roselin.S1, Benedict Michael Raj.L2
1MS. S. Jai Roselin., Research Scholar, Department of Mathematics, St.Josephโs college (Autonomous) affiliated to Bharathidasan University, Trichy.
2Dr. L. Benedict Michael Raj, Head and Associate Professor, St.Josephโs 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 ๐ฎ = (๐ฝ, ๐ฌ) and a finite set ๐ป of positive integers containing ๐, a ๐ป-coloring of ๐ฎ is a function ๐ โถ ๐ฝ (๐ฎ) โ ๐ + โช {๐} for all ๐ โ ๐ in ๐ฝ (๐ฎ) such that if ๐๐ โ ๐ฌ(๐ฎ) then |๐(๐) โ ๐(๐)| โ ๐ป. For a ๐ป-coloring ๐ of G, the f-span ๐๐๐ป ๐ (๐ฎ) is the maximum value of |๐(๐) โ ๐(๐)| over all pairs ๐, ๐ of vertices of ๐ฎ. The ๐ป-span ๐๐๐ป(๐ฎ) is the minimum ๐-span over all ๐ป-colorings f of ๐ฎ. The ๐-edge span ๐๐๐๐ป ๐ (๐ฎ) of a ๐ป-coloring is the maximum value of ๐ ๐ โ ๐ ๐ over all edges ๐๐ of ๐ฎ. The ๐ป-edge span ๐๐๐๐ป(๐ฎ) is the minimum ๐-edge span over all ๐ป-colorings f of ๐ฎ. It is known that ๐๐๐ป(๐ฏ) โค ๐๐๐ป(๐ฎ) and ๐๐๐๐ป(๐ฏ) โค e๐๐๐ป(๐ฎ) for every graph ๐ฎ. In this paper we classify which graphs containing a sub graph ๐ฏ such that ๐๐๐ป ๐ฏ < ๐๐๐ป(๐ฎ) and ๐๐๐๐ป(๐ฏ) < e๐๐๐ป(๐ฎ). Also we discuss the Mycielskian of ๐ป-coloring.
Keywords: T-coloring, T-span, T-edge SpanAMS Subject Classification 05C15
Scope of the Article: Classification