miscellaneous properties of middle graphs

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We have defined a graph M(G) as an intersection graph on V(G), and cailed ... X(G):{xr, x2t "'s xq}, where xt:(l)itt o;r), xz:(o;r,'i)'"'' xo:(u:" ttiq)' A graph. M(G) is def,ned as an intersection graph g(F) on V(G)' \'e" M(G):Q(F)' where r:{{u,}, {u,}, "', {oe}, ...
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MISCELLANEOUS PROPERTIES OF MIDDLE GRAPHS

BY 」lN

AKIYAMA,TAKASIil HAMADA AND IZUヽ

Printed from

TRU Mathemalrcs,

Vol. 10, 1974

1l YOSHIMURA

MISCELLANEOUS PROPERTIES OF MIDDLE GRAPHS* BY

JIN AKIYAMA, T,ITasHI HAMADA AND IZUMI YOSHIMURA

Abstract. Let G be an ordinary graph and v(G):{ut t)2t...r oj} be its point set. We have defined a graph M(G) as an intersection graph on V(G), and cailed it the middle graph of G, see f5l. The cycle multiplicity irt(G) of a graph G is the maximum number of line-disjoint cycles contained in G, see tll, t7l. Denote by [x] the greatest integer less than or equal to x, by {x} the integer equal to

-[-r],

by max deg G the greatest degree among the points of G, by p(z;i) the degree

of a point oi in G, by Go the subgraph induced by the points of odd degree in G, and by Ko the complete graph on p points. For any graph G, let ao(G), d{G), Bo(G), B,(G), oo(G):aoo(G), or(G):arr(G) denote its point covering number, its line covering number, its point independence number, its line independence number, its point-point covering number, its line-line covering number respectively, see [6]. We obtained the following resuits:

Tueonen pianar

if

1.

of a graph G is a forest, outerpianar, max deg GS1, max deg G) if p is odd,

4. For

the middle graph

graph Ke on

p points,

i.,'(M(Kt)):[]to* l)p(p-z)) if p is even. PRoor. Our proof divides into four ) 7 4 {)

cases according

to p=r (mod. 6)

(r:0,

1'

Crsr 1. p:1, 3 (mod. 6) Substitute 6k*1 or 6k*3 lor p in both sides of (4) in (3.1), then it is readily verified that the values ol both sides are equal. This shows that the equality ol the theorem is true.

Crse 2. p:5 (mod. 6), i.e., p:6k*5 (k:0, l' 2, "') The line set x(M(Kr)) ol M(K) is partitioned into a family ol line sets ol r(p(i' (i:1,2,..., p). In this case, it is easily derived from a theorem of Fort and Hedlund [2] that arbitrary cycle representation of each Kr{i) consists ol a lamily 0 whose elements are 3-cycles and only one 4-cycle (in this case I is a null set). We will show that lines of all these 4-cycles form 3-cycles and only one 5-cycle' Let Q:{6rtl li:1,2,..., p} and S be partitioned into a family of k sets N":{Kr("" Krtat, Kr\rt, Kr\6), Kr(), K}\