Labeling cacti with a condition at distance two

• Main Authors: S. K. Vaidya, D. D. Bantva
• Format: Article
• Language: English
• Published: Università degli Studi di Catania 2011-06-01
• Series: Le Matematiche
• Subjects: L(2,1)-labeling, λ-number, Cactus graph, Block cut point graph, One point union of cycles
• Online Access: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/775

<p class="MsoNormal" style="TEXT-ALIGN: justify; TEXT-INDENT: 0.5in; MARGIN: 0in 0in 0pt; mso-layout-grid-align: none"> </p><span style="font-size: small;"><span style="font-family: Times New Roman;"><p class="MsoNormal" style="text-align: justify; margin: 0in 0in 0pt; mso-layout-grid-align: none;">An <em style="mso-bidi-font-style: normal;">L(2,1)</em>-labeling of a graph <em style="mso-bidi-font-style: normal;">G</em> is a function <em style="mso-bidi-font-style: normal;">f</em> <span style="mso-spacerun: yes;"> </span>from the vertex set <em style="mso-bidi-font-style: normal;">V(G)</em> to the set of all nonnegative integers such that | <em style="mso-bidi-font-style: normal;">f (x)</em><em style="mso-bidi-font-style: normal;"><span style="font-family: Arial; mso-bidi-font-family: 'Times New Roman';">-</span> f (y)</em>| ≥ 2 if <em style="mso-bidi-font-style: normal;">d(x, y)</em> = 1 and | <em style="mso-bidi-font-style: normal;">f (x) - f (y)|</em> ≥ 1 if <em style="mso-bidi-font-style: normal;">d(x, y)</em> = 2. The <em style="mso-bidi-font-style: normal;">L(2,1)</em>-labeling number <em style="mso-bidi-font-style: normal;">λ(G)</em> of <em style="mso-bidi-font-style: normal;">G</em> is the smallest number <em style="mso-bidi-font-style: normal;">k</em> such that <em style="mso-bidi-font-style: normal;">G</em> has an <em style="mso-bidi-font-style: normal;">L(2,1)</em>-labeling with max{<em style="mso-bidi-font-style: normal;">f (v) : v </em><span style="font-family: Symbol; mso-bidi-font-family: Symbol;">Î</span><em style="mso-bidi-font-style: normal;"> V(G)</em>} = <em style="mso-bidi-font-style: normal;">k</em>. In this paper we present a graph family which has λ-number Δ+1 or Δ+2.</p></span></span>