The sum number of the cocktail party graph
01 Jan 1998-
TL;DR: In this paper, the complete n-partite graph G = H m;n was considered and an optimal labeling of the vertices of G by distinct positive integers was given, where the vertex u and v are adjacent if and only if there exists a vertex u + v. This was the first known graph with this property.
Abstract: A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a finite number of isolated vertices to it will always yield a sum graph, and the sum number oe(G) of G is the smallest number of isolated vertices that will achieve this result. A labelling that realizes G + K oe(G) as a sum graph is said to be optimal. In this paper we consider G = H m;n , the complete n-partite graph on n 2 sets of m 2 nonadjacent vertices. We give an optimal labelling to show that oe(H 2;n ) = 4n \Gamma 5, and in the general case we give constructive proofs that oe(H m;n ) 2 \Omega\Gamma mn) and oe(H m;n ) 2 O(mn 2 ). We conjecture that oe(H m;n ) is asymptotically greater than mn, the cardinality of the vertex set; if so, then H m;n is the first known graph with this property. We also provide for the first time an optimal labelling of the complete bipatite graph Kmn whose smallest label is 1.
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Journal Article•
TL;DR: In this survey I have collected everything I could find on graph labelings techniques that have appeared in journals that are not widely available.
Abstract: A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.
2,367 citations
Cites background from "The sum number of the cocktail part..."
...Miller, Ryan, and Smyth [873] prove that the complete n-partite graph on n sets of 2 nonadjacent vertices has sum number 4n−5 and obtain upper and lower bounds on the complete n-partite graph on n sets ofm nonadjacent vertices....
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TL;DR: An optimal sum labelling scheme is provided for the generalised friendship graph, also known as the flower, and it is shown that its sum number is 2.
14 citations
Cites background from "The sum number of the cocktail part..."
...Tree Tn, n 2 1 [2] Cycle Cn, n 3, n = 4 2 [3] 4-Cycle C4 3 [3] Even wheel Wn, n 4, n even n2 + 2 [5,8] Odd wheel Wn, n 5, n odd n [5,8] Complete graph Kn, n 4 2n− 3 [1] Cocktail party graph H2,n, n 2 4n− 5 [9] Complete bipartite graph ⌈ k n−1 2 + m k−1 ⌉ [4,6,10,12]...
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TL;DR: It is proved that the sum number of a hypertree (≔ connected, non-trivial and cycle-free hypergraph) is equal to 1, if a certain condition for the cardinalities of the edges is fulfilled.
10 citations
Cites background from "The sum number of the cocktail part..."
...Secondly, there are a lot of results on the sum number of certain classes of graphs: for complete graphs [3,6], for complete bipartite graphs [3,16], for trees [8], for paths and caterpillars [13], for wheels [17,21], for cocktail party graphs [20], for graphs of small sum number [23] and for arbitrary graphs as well as for other classes of graphs [10]....
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TL;DR: Giving a special labelling algorithm, it is proved that cacti with a girth of at least 6 are difference graphs, too.
Abstract: A graph G is a difference graph iff there exists S ⊂ IN such that G is isomorphic to the graph DG(S) = (V,E), where V = S and E = {{i, j} : i, j ∈ V ∧ |i− j| ∈ V }. It is known that trees, cycles, complete graphs, the complete bipartite graphs Kn,n and Kn,n−1, pyramids and n-sided prisms (n ≥ 4) are difference graphs (cf. [4]). Giving a special labelling algorithm, we prove that cacti with a girth of at least 6 are difference graphs, too.
8 citations
TL;DR: Graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions as mentioned in this paper , and graph labeling techniques have been studied in over 3000 papers.
Abstract: A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid-1960s. In the intervening years over 200 graph labelings techniques have been studied in over 3000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey, I have collected everything I could find on graph labeling. For the convenience of the reader, the survey includes a detailed table of contents and index. This edition has 267 new references that are identified with the reference number and the word "new" in the right margin.
6 citations
References
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Journal Article•
TL;DR: For even n 4 and odd n 5, it was shown in this paper that oe(W n ) = n 2 + 2 for wheels W n of order n+1 and size m = 2n, where m is the size of the graph.
Abstract: A simple undirected graph G is called a sum graph if there exists a labelling L of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w whose label L(w) = L(u) +L(v). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The sum number oe(H) of a connected graph H is the least number r of isolated vertices K r such that G = H+K r is a sum graph. It is clear that if H is of size m, then oe(H) m. Recently Hartsfield and Smyth showed that for wheels W n of order n+1 and size m = 2n, oe(W n ) 2 Theta(m); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper we refine these results to show that for even n 4, oe(W n ) = n=2 + 2, while for odd n 5 we disprove a conjecture of Hartsfield and Smyth by showing that oe(W n ) = n. Labellings are given that achieve these minima.
27 citations
"The sum number of the cocktail part..." refers background in this paper
...In fact, the only known class of graphs G that even achieves as much as (G) 2 (jEj) is the class of wheels Wn with n spokes: as shown in [6] and [7], (Wn) = n=2 + 2 for n even; = n for n odd....
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01 Jan 1992
26 citations
TL;DR: In this paper, it was shown that for wheels Wn of (sufficiently large) order n + 1 and size m = 2n, σ(Wn) = n/2 + 3 if n is even and n ⩽ σ (Wn), ⌈ n + 2 if n was odd.
26 citations
"The sum number of the cocktail part..." refers background in this paper
...In fact, the only known class of graphs G that even achieves as much as (G) 2 (jEj) is the class of wheels Wn with n spokes: as shown in [6] and [7], (Wn) = n=2 + 2 for n even; = n for n odd....
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