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Solutions of Some L(2, 1)-Coloring Related Open Problems

Nibedita MandalPratima Panigrahi — 2016

Discussiones Mathematicae Graph Theory

An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring f : V (G) → Z+ ∪ {0} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of the graph is...

Some new classes of graceful Lobsters obtained from diameter four trees

Debdas MishraPratima Panigrahi — 2010

Mathematica Bohemica

We observe that a lobster with diameter at least five has a unique path H = x 0 , x 1 , ... , x m with the property that besides the adjacencies in H both x 0 and x m are adjacent to the centers of at least one K 1 , s , where s > 0 , and each x i , 1 i m - 1 , is adjacent at most to the centers of some K 1 , s , where s 0 . This path H is called the central path of the lobster. We call K 1 , s an even branch if s is nonzero even, an odd branch if s is odd and a pendant branch if s = 0 . In the existing literature only some specific classes of lobsters have been found...

Nearly antipodal chromatic number a c ' ( P n ) of the path P n

Srinivasa Rao KolaPratima Panigrahi — 2009

Mathematica Bohemica

Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number a c ' ( P n ) as n - 2 2 + 2 for n 9 and have found the exact value of a c ' ( P n ) for n = 5 , 6 , 7 , 8 . Here we determine the exact values of a c ' ( P n ) for n 8 . They are 2 p 2 - 6 p + 8 for n = 2 p and 2 p 2 - 4 p + 6 for n = 2 p + 1 . The exact value of the radio antipodal number a c ( P n ) for the path P n of order n has been determined by Khennoufa and Togni in 2005 as 2 p 2 - 2 p + 3 for n = 2 p + 1 and 2 p 2 - 4 p + 5 for n = 2 p . Although the value of a c ( P n ) determined there is correct, we found a mistake in the proof of the lower bound when n = 2 p (Theorem 6 ). However,...

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