𝔖 Bobbio Scriptorium
✦   LIBER   ✦

The -conjecture for -labelings is true for total graphs

✍ Scribed by Ziming Duan; Pingli Lv; Lianying Miao; Zhengke Miao; Cuiqi Wang

Publisher
Elsevier Science
Year
2011
Tongue
English
Weight
210 KB
Volume
24
Category
Article
ISSN
0893-9659

No coin nor oath required. For personal study only.

✦ Synopsis

An L(2, 1)-labeling of a graph G is defined as a function f from the vertex set V (G) into the nonnegative integers such that for any two vertices x, y, |f

Griggs and Yeh conjectured that Ξ» 2,1 (G) ≀ βˆ† 2 for any simple graph with maximum degree βˆ† β‰₯ 2. In this paper, we consider the total graph T (G) of a graph G and derive its upper bound of Ξ» 2,1 (T (G)). Shao, Yeh and Zhang had proved that Ξ» 2,1 (T (G))

which shows that the conjecture of Griggs and Yeh is true for the total graph. In addition, we obtain the exact value of Ξ» 2,1 (T (K m,n )) for the total graph of a complete bipartite graph K m,n with m β‰₯ n β‰₯ 1.

πŸ“œ SIMILAR VOLUMES

The McKay Conjecture Is True for the Spo
The McKay Conjecture Is True for the Sporadic Simple Groups
✍ Robert A. Wilson πŸ“‚ Article πŸ“… 1998 πŸ› Elsevier Science 🌐 English βš– 146 KB

The McKay conjecture states that the number of irreducible complex characters of a group G that have degree prime to p is equal to the same number for the Sylow p-normalizer in G. We verify this conjecture for the 26 sporadic simple groups.

The reconstruction conjecture is true if
The reconstruction conjecture is true if all 2-connected graphs are reconstructible
✍ Yang Yongzhi πŸ“‚ Article πŸ“… 1988 πŸ› John Wiley and Sons 🌐 English βš– 344 KB

It is shown that the Reconstruction Conjecture is true for all finite graphs if it is true for the 2-connected ones. We shall, for the most part, use the terminology of [2] and [ 4 ] . Graphs will be finite, simple, and undirected. Let G be a graph and u E V(G). Denote by d(u) the degree of u in G