✦ LIBER ✦

Subdividing a Graph Toward a Unit-distance Graph in the Plane

✍ Scribed by Severino V. Gervacio; Hiroshi Maehara

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 150 KB
Volume: 21
Category: Article
ISSN: 0195-6698
DOI: 10.1006/eujc.1999.0348

No coin nor oath required. For personal study only.

✦ Synopsis

The subdivision number of a graph G is defined to be the minimum number of extra vertices inserted into the edges of G to make it isomorphic to a unit-distance graph in the plane. Let t (n) denote the maximum number of edges of a C 4 -free graph on n vertices. It is proved that the subdivision number of K n lies between n(n -1)/2 -t (n) and (n -2)(n -3)/2 + 2, and that of K (m, n) equals (m -1)(nm) for n ≥ m(m -1).

📜 SIMILAR VOLUMES

Every Graph Is an Integral Distance Grap

Every Graph Is an Integral Distance Graph in the Plane

✍ Hiroshi Maehara; Katsuhiro Ota; Norihide Tokushige 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 290 KB

We prove that every finite simple graph can be drawn in the plane so that any two vertices have an integral distance if and only if they are adjacent. The proof is constructive.

Distance-regular Subgraphs in a Distance

Distance-regular Subgraphs in a Distance-regular Graph, III

✍ Akira Hiraki 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 254 KB

Let ⌫ be a distance-regular graph with l (1 , a 1 , b 1 ) ϭ 1 and c s ϩ 1 ϭ 1 for some positive integer s . We show the existence of a certain distance-regular graph of diameter s , containing given two vertices at distance s , as a subgraph in ⌫ .

Distance-regular Subgraphs in a Distance

Distance-regular Subgraphs in a Distance-regular Graph, IV

✍ Akira Hiraki 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 280 KB

Let ⌫ be a distance-regular graph with a 1 Ͼ 0 , r ϭ max ͕ j 3 ( c j , a j , b j ) ϭ ( c 1 , a 1 , b 1 ) ͖ у 2 and a i ϭ a 1 c i , for 1 р i р 2 r . Take any u and in ⌫ at distance r ϩ 1 . We show that there exists a collinearity graph of a generalized 2( r ϩ 1)-gon of order ( a 1 ϩ 1 , c r ϩ 1 Ϫ 1)

Distance-regular Subgraphs in a Distance

Distance-regular Subgraphs in a Distance-regular Graph, V

✍ A. Hiraki 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 186 KB

Distance-regular Subgraphs in a Distance

Distance-regular Subgraphs in a Distance-regular Graph, VI

✍ A. Hiraki 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 202 KB

In this paper we give a sufficient condition for the existence of a strongly closed subgraph which is (c q + a q )-regular of diameter q containing a given pair of vertices at distance q in a distance-regular graph. Moreover we show that a distance-regular graph with r = max{ j | (c j , a j , b j )

The average Steiner distance of a graph

✍ Dankelmann, Peter; Oellermann, Ortrud R.; Swart, Henda C. 📂 Article 📅 1996 🏛 John Wiley and Sons 🌐 English ⚖ 384 KB 👁 2 views

The average distance p(G) of a graph G is the average among the distances between all pairs of vertices in G. For n 2 2, the average Steiner n-distance ,4G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, pu,