IJPAM: Volume 80, No. 1 (2012)

ON $k-$GRACEFULNESS OF $r-$CROWNS
FOR COMPLETE BIPARTITE GRAPHS

Deligen$^1$, Lingqi Zhao$^2$, Jirimutu$^3$
$^{1,3}$College of Mathematics
Inner Mongolian University for Nationalities
Tongliao, 028043, P.R. CHINA
$^2$College of Computer Science and Technology
Inner Mongolian University for Nationalities
Tongliao, 028043, P.R. CHINA


Abstract. Let $I_r(K_{m,n})$ denote a $r-$ crown of a complete bipartite graph $K_{m,n}$ obtained by adding $r$ hanged edges to each vertex of $K_{m,n}$. Ma kejie conjectured that 1-crown of complete bipartite graph $K_{m,n}$ $(m \leq n)$ is $k-$ graceful graph for $k \geq 2$. The conjecture has been shown true when $m=1,2,3,4$ for arbitrary $n \geq m$ and $r \geq 2$. In this paper we discuss the $k-$gracefulness of $r-$crown $I_r(K_{m,n})$ $(m \leq n,r \geq 2)$ for complete bipartite graph $K_{m,n}$ and prove the conjecture when $m=5$, for arbitrary $n \geq m$ and $r \geq 2$.

Received: March 12, 2012

AMS Subject Classification: 05C65

Key Words and Phrases: complete bipartite graph, graceful graph, $k-$graceful graph

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 1