Investigation on Interval Edge-Colorings of Graphs

 

Investigation on Interval Edge‐Colorings of 

Graphs 

Armen Asratian and R. R. Kamalian

The self-archived postprint version of this journal article is available at Linköping

University Institutional Repository (DiVA):

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-143759

  

  

N.B.: When citing this work, cite the original publication.

Asratian, A., Kamalian, R. R., (1994), Investigation on Interval Edge-Colorings of Graphs, Journal of combinatorial theory. Series B (Print), 62(1), 34-43. https://doi.org/10.1006/jctb.1994.1053

Original publication available at:

https://doi.org/10.1006/jctb.1994.1053

Copyright: Elsevier

http://www.elsevier.com/

 

 

 

 

 

JOURNAL OF COMBINATORIAL THEORY, Series B 62, 34--43 (1994)

Investigation on Interval Edge-Colorings of Graphs

A. S. AsRATIAN

Department of Mathematical Cybernetics, Yerevan State University, Yerevan, 375049, Republic of Armenia

AND

R. R. KAMALIAN

Computing Centre, Academy of Sciences of the Republic of Armenia, Yerevan, 375014, Republic of Armenia

Received February 8, 1991

An edge-coloring of a simple graph G with colors !, 2, ... , t is called an interval t-coloring [3] if at least one edge of G is colored by color i, i = 1, ... , t and the edges incident with each vertex x are colored by dG(x) consecutive colors, where dG(x)

is the degree of the vertex x. In this paper we investigate some properties of interval colorings and their variations. It is proved, in particular, that if a simple graph

G = ( V, E) without triangles has an interval t-coloring, then t,;; I VI - 1. 1;, 1994 Academic Press, Jnc.

1. INTRODUCTION

All

graphs considered in this paper are undirected and have no loops or multiple edges. V(G) and E(G) denote the sets of vertices and edges of a graph G, respectively. The degree of a vertex x in G is denoted by dG(x),

and the diameter of G is denoted by d(G). A t-coloring of a graph G is a function f: £( G)---+ { 1, ... ,

t},

such that f(e) i= f ( e') for any pair of adjacent edges e and e'. If eEE(G) and f(e)=i then e is said to be colored by color i. The chromatic index x'( G) is the least value of t for which a t-coloring of G exists. A well-known theorem of V. G. Vizing [ 17] states that Ll(G)<x'(G)<Ll(G)+l where Ll(G) denotes the maximum degree of G. If x'(G)

=

Ll(G) then G is said to be Class 1 and otherwise G is Class 2. The problem of deciding whether or not a regular graph is Class 1 was shown by I. Holyer [10] to be %&'-complete (see also [15]).

Let R i:;;. V( G). An interval (R, t )-coloring of G is a t-coloring of G such

that at least one edge of G is colored by color i, i

=

1, ... , t, and the edges incident with each x ER are colored by dG(x) consecutive colors. An inter-val t-coloring of G is an interval ( V( G ), t )-coloring of G. Let fil

=

U,;;,

1 fil,,

where fil, is the set of graphs having an interval t-coloring.

34 0095-8956/94 S6.00

Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved.

A bipartite graph G with the partition V1(G)u V2(G) is denoted by

=(V1(G), V2(G), E(G)).

An interval (V1(G),t)-coloring of a bipartite graph G=

(V1(G), V2(G), E(G)) is called a Vi(G)-sided interval t-coloring of G. Let w 1 ( G) be the least value of t for which a Vi( G)-sided interval t-coloring of a bipartite graph G exists.

Some results on interval colorings were obtained in [2, 3, 7, 11, 12]. Interval vertex-colorings were considered in [14, 19].

In this article we obtained the following results:

(1 ) If G E fil then

x' (

G) = A ( G ).

(2) If GE fil, then t ~ 2 I V(G)l -1; if, moreover, G has no triangle then t ~ IV( G)I - 1.

(3) If G is bipartite and GEfil, then t~A(G)·d(G)-d(G)+ 1.

(4) If G

=

(V1 (G), V2(G), E(G)) then for every t, H' 1 (G) ~ t ~ IE(G)I,

there exists a V1 (G)-sided interval t-coloring of G.

Note that some timetable problems with compactness requirements (i.e., the lectures of each teacher and (or) each group have to be scheduled at consecutive periods) may be formulated as problems of interval edge-colorings of bipartite graphs. Other applications of edge-colorings of bipartite graphs were considered in [l, 4, 5, 18].

Our notation and terminology follows C. Berge [ 1] and F. Harary [9].

2. RESULTS PROPOSITION 1.

If

GE fil then G is Class 1.

Proof Consider an interval t-coloring f of G. Let E₁

=

{eEE(G)/f(e)=J (mod(A(G)))}, j= 1, ... , A(G). It is not difficult to see, that for j

=

1, ... , LI( G), Ei is a matching in G. Therefore coloring the edges of EJ by color j for j= 1, ... , A(G) gives a A(G)-coloring of G. So

x'(

G) = A( G). The proof is complete.

COROLLARY 1. Jf a graph G is Class 2 then G ~ fil.

COROLLARY 2.

If

G is an r-regular graph, then GE 2l

if!

x'(G) = r.

From Corollary 2, in particular, it follows that the set 2l contains all regular bipartite graphs and all complete graphs with an even number of vertices. Furthermore, from Corollary 2 and the result of I. Holyer [10] it follows that the problem "Does for a given r-regular graph an interval r-coloring exist or not?" is A'" EP- complete [ 13].

36 ASRA TIAN AND KAMA LIAN

LEMMA 1. The complete bipartite graph Km, n has an interval (m

+

n - 1 )-coloring.

Proof Let V1(Km.n)= {x1 , ... , xm} and V2(Km,n)= {Y1, ... , Yn}. Define a coloringf by settingf((x;, y_{1 ))}

=

i

+

j -1, i = 1, ... , m,j = 1, ... , n. It is easy to see, that f is an interval (m

+

n - 1 )-coloring of Km, n- The proof is complete.

PROPOSITION 2. The set fil contains all trees and all complete bipartite graphs.

Proof Let G be a tree. By induction on the number of vertices one can prove that G has an interval LI ( G )-coloring. Therefore GE fil.

Now let G

=

Km, n· By Lemma l Km. n has an interval (m

+

n - 1 )-coloring. Therefore K,,,, n E fil. The proof is complete.

In [ 12], in the case when G is a tree or a complete bipartite graph, all values of t were found for which an interval t-coloring of G exists.

LEMMA 2. Let f be a t-coloring of a graph G and let H be a connected subgraph of G such that the edges of H incident with each vertex x E V(H) are colored by consecutive colors. If S is the set of co/ors used in H then H has an interval I Sl-cowring. Moreover, H has an interval t-coloring if

{1,t}sS.

Proof Let t 1 = min,E E(Hl f(e) and t2 = maxeE E(Hl f(e ). Since His

con-nected then there exists a simple path P= (v0 , e1 , v1 , ... , vk_1 , ek, vd in H,

where e;=(v;_1,v;), i=1, ... ,k, and f(ei)=t2 , f(ed=t1 • If f(e;)#q,

i = 1, ... , k, for some color q ES then t 1 < q < t 2 and there exists an integer

i0 , 1,:;;_i0<k, such that f(e;0)>q and f(e 1+;0)<q. Therefore, there is an edge of H colored q which is incident with X;

0, So for any q, t1 ,:;;_q,:;;_12

there is an edge in H colored q. Therefore ISI

=

t_{2 -} t1

+

1. Now consider an ISl-coloring/1 of H, where/1(e)=f(e)-t1

+

1 for each edge eEE(H). Clearly, /1 is an interval I Sj-coloring of H. If t 1 = 1 and t 2 = t then I SJ

=

t, and /1 is an interval t-coloring of H. The proof is complete.

Letfil)={GEfil,/IV(G)l~t}, t~l.

PROPOSITION 3. fil: #

0

for any t ~ 4.

Proof If t = 2n consider a (2n - 1 )-coloring f of the complete graph

K2n. Let V(K2n)= {v1 , ... , v2n}. Without loss of generality we suppose thatf((v1,v2))=1 andf((v1,v3))=2. Now we can define an interval

2n-coloring /1 of K2n, where /1 (( v 1 , v2 ))

=

2n and /1 ( e) = f( e) for each edge e-f,. (v_{1 ,} v_{2 ).} Therefore K2nEfil:. (It is shown in [11] that K 2n has an

interval (2n - 1

+

Llog2(2n - 1 )J)-coloring, where

LcxJ

is the integral part

of a number !Y. ).

Let t = 2n

+

1, n ~ 2. Consider a graph Fn with V(Fn) = { v1 , ... , v2n,

V 2n+ i} and E(Fn) = E(K2n) U { (v1 , v 2n+ i), (v 3, v2n+ 1) }. Using the interval

2n-coloring/1 of K2n we define an interval (2n

+

1 )-coloring/2 of Fn, where

f2((v1, V2n+1))=2, f2((v3, V2n+d)= 1, and f2((v;, v_1))=f1((v;, v₁))+ 1,

1 :::; i < j :::;; 2n. So Fn E fil

1n

_{+ 1. The proof is complete.}

Now we shall show that for any t ~ 1 the set

fil:

contains no graph without triangles.

THEOREM 1. If a graph G has no triangle and GE fil, then t:::;;

I

V( G)

I -

1.

Proof Suppose that the theorem is false. Let t be the least integer for which the set

m::

contains graphs without triangles and let G be a graph from fil) with the minimum number of edges. Without loss of generality we can assume that G is a connected graph with /E(G)I > 1. For uE V(G) let

N(u)= {vE V(G)/(u, v)EE(G)}.

Consider an interval t-coloring f of G. We denote by JI the set of those simple paths which start with an edge colored t and finish with an edge colored 1. To each path PEA with a sequence of edges e1 , ... ,eq (q~2) there corresponds the sequence f(P) of colors of its edges, where f(P) =

(f(ei), ... , f(eq)). Now we will show that there exists a P0 in JI for which

f(P0 ) is decreasing.

Let f(e')=t, e'=(x0,xi), and dc(x1)"?:dc(x0 ). Since JE(G)J>l then dG(x _{1 )} ~ 2. We shall construct a sequence X of vertices of G in the following way:

Step 1. X :=

{x

0 , xi}.

Step 2. Let X; be the last constructed vertex in the sequence X. If N(x;)\X=0 or f((x;,y))>f((x;_1,x;)) for each yEN(x;)\X then the construction of X is completed. Otherwise we find the vertex x; + 1 in

N(x;) \X for which f((x;, X;+ 1 )) = minyE Ntx,J • x f((x;, y) ). Add X;+ 1 to X

and repeat Step 2.

Suppose that X has been already constructed and X= {x0 , x 1, ... , xk}.

Obviously X defines a simple path P0= (x0 , e 1, x,, ... , xk_,, ek> xk), where

e;= (x;_ 1, x;), i= 1, ... , k, andf(ed < t. Define a graph Hin the following

way: if dG(xk)= 1 then H=G-xk; otherwise H=G-ek.

Let us show that H is connected. Assume that it is not connected. Then H = G-ek. We denote by F₁ and F₂ the connected components of H, where xk-I E V(Fi), xk E _V(F2). Let H 1 and _{H 2} be the subgraphs of G

induced by the sets V(F₁)u {xd and V(F₂)u {xk_i}, respectively. Denote by S; the subset of colors used in H;, i

=

1, 2. The edges of H; incident with

38 ASRA TIAN AND KAMALIAN

each vertex x E V(H;) are colored by consecutive colors. Therefore, by Lemma 2, H; has an interval IS;l-coloring, i= 1, 2. Since IE(H;)I < IE(G)I then IS;I ~ IV(H;)l-1, i= 1, 2. Finally, t _{= IS1}

us

21 = ISil + IS2I

-IS1 n S2I ~ IS1I + IS2I - l ~ I V(H1)I + I V(H2)1-3 = I V(G)l -1.

This inequality contradicts the choice of G. Consequently H is connected.

Let us show that

/(ed

= 1. Assume that f(ed > 1. The edges of H

incident with each vertex x E V(H) are colored by consecutive colors. Since l <f(ek) < t then there are edges in H colored by l and t. Therefore, by Lemma 2, H has an interval t-coloring. We have HE fil1 and

t ~ IV(G)I ~ IV(H)I. This inequality contradicts the choice of G because H

has no triangle and IE(H)I < IE(G)I. Therefore f(ed = 1. So we have constructed a path P0 EA for which the sequence /(P0 ) is decreasing.

Let {) be the set of all the shortest paths P in j { with decreasing f(P).

We denote by k the length of paths of

e.

Now we need subsets {) 1, ... , {)k defined in the following way:

e

_{1 = {) and}

{); is a subset of paths from {);_ 1 with the greatest color of the ith edge, i=2, ... , k.

We choose some path P1=(x0,e1,x1 , ... ,xk-i,ek>xd from {)k· Let

A(i)={yEN(x;)/f(e;+d < f((x;,y)) < f(e;)}, i=l, ... ,k-1. Evidently,

I

A ( i)

I

= /( e;) - /( e; +

d -

1, i = 1, ... , k - 1.

Let us show that A(i)n{x0 , ... ,xd=0, i=l, ... ,k-1. Assume that

there exist i0

,J

0 for which X;0EA(j0 ) or x10EA(i0 ). Let us define a path P'. If i0 # 0, Jo# k then P' = (x0 , e1 , x 1 , ... , X;0, (x;0, x10), x10, ... , xd. If

i0 = 0 then P' = (x 1, e1 , x0 , (x0 , x10 ), x10, ... , xd. If Jo= k then

P'=(x0,e1,x1 , ... ,x;0, (x;0,xk), xbek>xk_1 ). In all cases f(P') is

decreasing and P' is shorter than P_I which contradicts the choice of P _{1 •}

Now we shall show that A (i) n A (j) = 0, l ~ i <

J

~ k - 1. Suppose that there exist i0

,J

0 , l ~i0

<J

0~k- l for which A(i0)nA(j0)#0. Since G

has no triangle thenJ0- i0~2. Let vEA(i0)nA(j0 ). Consider a new path

P" = (x0 , e1 , x1 , ... , X;0 , (x;0, v), v, (v, x10 ), x10, ... , xk_ 1, ek> xd. Clearly f(P") is decreasing. If Jo - i0 ~ 3 then P" is shorter than P 1. If Jo - i0 = 2 we have

f((x;₀, v)) >

/(e

₁ + ;₀). So in both cases we have a contradiction to the choice

of P1 •

Now we can conclude that

k-1 k-1

IV(G)l~k+l+

L

IA(i)l=k+l+

L

(f(e;)-f(e;+i)-1)

i= l = k + I + t - I - (k - 1) = 1 + t.

It contradicts the choice of G. The proof is complete. From Theorem l we conclude:

From Lemma 1 it follows that the bounds of Theorem 1 and Corollary 3 are sharp for Km, n because

I

V(Km,

n)I

=

m

+

n.

S. V. Sevastianov showed [16] that the problem "Whether a given biparite graph belongs to the set 21 or not?" is %&'-complete.

PRorosmoN 4. If GE 21, then t ~ 2 IV(G)I - 1.

Proof Let V(G)

=

{x

1 , ...

,xn}.

We define a bipartite graph

G* = _(V1(G*), V₂(G*), E(G*)) in the following way: V₁(G*)

=

{y

1 , ••• , Yn},

V2(G*)={z1 , ••• ,zn}, E(G*)={(Y;,z1 ), (z;,Y)/(x;,X)EE(G), l~i~n, l~j~n}.

Consider an interval t-coloring f of G. It is easy to see, that f induces an interval t-coloring f* of G*, where f*((y;, zi))

=

f*((z;, YJ))

=

f( (x;, xJ)) for each edge (x;, x_{1 )} of G. Since G* is a bipartite graph and

G* E 21, then, by Corollary 3, t ~ I V(G*)I -1

=

2 I V(G)I - 1. The proof is complete.

Let .1 ;( G)

=

max XE v,fGJ ddx ), i

=

l, 2, for a bipartite graph

G

=

(VI (G), V2(G), E(G)).

THEOREM 2. If G is a bipartite graph from 21, and .1₁(G)~.1₂(G), then

t~ { i(d(G)

+

1) .11 (G)

+

i(d(G)- 1) .12(G)-d(G)

+

1 if d( G) is odd,

!

d(G)(.1 I (G)

+

d2(G))- d(G)

+

1 if d( G) is even.

Proof Consider an interval t-coloring f of G. Let f(e)

=

1, /(e')

=

t, e=(u1,u2 ), e'=(v1,v2 ). Let PiJ be the shortest path joining u; with vJ,

i= l, 2, j= 1, 2. Let P0 be the shortest path among PiJ, i= 1, 2, j= 1, 2. Without loss of generality we can suppose that P₀ joins u1 with v1 • Now consider the path P21 • If the length of P21 is equal to the length of P0 then

G has an odd cycle; but this is impossible because G is bipartite. Conse-quently P21 is longer than P0 • Therefore the length of P0 is no greater than

d(G)-1. Let P0

=

(x1,e1,x2,e2 , x3 , ••• ,x;, e;,X;+1 , ••• ,xk> ek> xk+d, where x1=u1 , xk+1=v1 • Clearly

f(ei)~dG(xi),

40 ASRA TIAN AND KAMAUAN From these inequalities we obtain

k+ l k+ l

t~

L

da(x;)-k= 1

+

L

(da(x;)-1).

i= l i = l

Now we define some auxiliary numerical sequences, taking into account that k~d(G)-1: 1. for i

=

l, ... , k + l,

{'1

1(G) if X;E Vi(G), e;= L12(G) _{if X;E V2(G);} 2. for i= 1, ... , k +I, ... , d(G), if l~i~k+l, (J;

r

e: if k +I< i~ d(G) and 82 if k+ 1 < i~ d(G) and 3. fori=l, ... ,k+l, ... ,d(G), { L11(G) a,= L12(G) Using these sequences we obtain

if i is odd, if i is even. i is odd, i is even; k+ l k+ l d(G) /~ 1 +

L

(dc(x;)-1)~ 1 +

L

(e;-1)~ 1 +

L

((J;-1) i= 1 i=l d(G) d(G) ~ 1

+

L

(a; - 1)

=

I

a; - d( G)

+

1. i=l i=l

It is easy to note that

d(G) {1(d(G) + 1) j I (G) + !(d(G)- 1) L12(G)

L

a;= if d(G) is odd,

i=l !d(G)(J₁(G)+J 2(G)) if d(G) is even This completes the proof.

COROLLARY 4. If G is a bipartite graph and GE 21, then t ~ d( G) · J ( G) - d( G) + 1.

It follows from Lemma 1 that the bound of Corollary 4 is sharp for Kn, n

when n~2 since in this case d(Kn.n)=2, J(Kn.n)=n.

Let us compare the inequalities of Corollary 3 and Corollary 4. If

A(Pn)

=

2, IV(Pn)I

=

n, and IV(Pn)I - 1

=

n - 1::::;; d(Pn) · Ll(Pn)- d(Pn)

+

1

=

n, so in this case the inequality of Corollary 3 is better than that of Corollary 4.

If G=Qn, where Qn is then-cube [4] then d(Qn)=n=A(Qn),

I

V(Qn)I

=

2n, d(Qn) · Ll(Qn)- d(Qn)

+

1

=

n2

- n

+

1

<

I

V(Qn)I - 1

=

2n - 1,

so in this case the inequality of Corollary 4 is better than that of Corollary 3.

With an argument similar to that of Theorem 2, we can derive some further results.

PROPOSITION 5.

If

GE fil, and G has an odd cycle then t<(d(G)+ 1)-A(G)-d(G).

COROLLARY 5.

If

GEfil, and dG(u)+dG(v)~ IV(G)l-1 for every pair

of nonadjacent vertices u and v then t

<

3 · A(G)- 2.

THEOREM 3. Let G= (Vi(G), V2(G), E(G)) be a bipartite graph.

Then for every t satisfying the inequality w1 (G)

<

t

<

IE(G)I there exists a

Vi(G)-sided interval t-coloring of G.

Proof The proof is by induction on IV1(G)I. When !Vi(G)I

=

I the

theorem is clear.

Suppose that the theorem holds for all graphs G' with JVi(G')I =p. Suppose that )V,(G)I =p+ 1. Assume there exists a V,(G)-sided interval t-coloring of G, where wi(G)

<

t < IE(G)I. Consider the intersection of V1{G) with the set of endpoints of edges colored t and choose from this

intersection a vertex x _I with the smallest degree. Evidently there exists an edge e i colored by color t

+

1 - dG(x i ), which is incident with x i ·

Case 1. There exists an edge colored by color t

+

1 - ddx i) disjoint from e1 • In this case we recolor ei by color t+ 1 and obtain a Vi(G)-sided interval ( t

+

1 )-coloring of G.

Case 2. e1 is the only edge colored by color t+ 1-ddxi) and sis the maximum color by which more than one edge is colored. Clearly, 1 <s<t< IE(G)I.

Case 2a. t

+

1 -dG(xi) < s < t. Recolor the edges colored by i, i

=

t

+

1 - dG(x i), ... ,

s,

by color i

+

t

-s.

Similarly, recolor the edges colored by i, i

=

s

+

1, ... , t, by color

i-s+t-dG(xi). Evidently, this recoloring gives a new Vi ( G)-sided interval t-coloring of G. Clearly, in this

new coloring the intersection of Vi (G) with the set of endpoints of edges colored t contains more than one ver-tex. Choose from this intersection a vertex x 2 with the

42 ASRA TIAN AND KAMALIAN

smallest degree. Now among the edges which are incident with x₂ we choose the edge with the smallest color and recolor it by color t

+

1. It is easy to see, that we obtain a VI G)-sided interval (t

+

1 )-coloring of G.

Case 2b. 1 ~ s < t

+

1 - dG(xi). We delete x1 from G and obtain a graph G' with a V1 (G')-sided interval (t-dG(x1 ))-color-ing. Clearly, IE(G')I

=

IE(G)l-dG(xi) and t-dG(xi)<

IE(G')I. According to the induction hypothesis there exists a V1(G')-sided interval (t+ 1-ddxi))-coloring

of G'. Now we color the edges incident with x1 using the

colors t

+

2 - do(x i), ... , t

+

l. This gives a V1 ( G)-sided interval (t

+

1 )-coloring of G.

The proof now follows by induction.

Clearly, w₁(G)?A(G) for each bipartite graph G.

PROPOSITION 6. For any positive integer p there exists a bipartite graph G such that w1(G)?L1(G)+p.

Proof Let us choose a positive integer n satisfying the condition

n-2?p. Consider a complete bipartite graph Kn+i,n, where V1(Kn+i,n)

=

{x1, ... , Xn+

1},

V2(Kn+ l,n) =

{Y1, ... ,

Yn}. Now we delete the subset of edges { (x;, y;)/l ~ i ~ n} from Kn+ 1, n and denote by G the remaining graph. Clearly, A(G)

=

n.

Let us show that w 1 ( G)? 2n - 2. Assume that w 1 ( G) ~ 2n - 3. It is not

difficult to note that if we have at most 2n - 3 colors, then at each vertex

x of V1 ( G) we can not avoid the color n - l because do(x)? n - l for any x E Vi( G). Since

IV

1 ( G)I = n

+

1 there must be n

+

1 different edges in G colored by col or n - l, but it is impossible because I V 2 ( G)I

=

n < n

+

1. The proof is complete.

PROPOSITION 7 [2]. Let G= (V1(G), V2(G), E(G)) be a bipartite graph.

If dG(x)?do(y)for each edge (x, y) with XE V1(G), yE V2(G), then G has a A(G)-coloring such that the edges incident with every vertex XE V1(G) are colored by co/ors 1, ... , dG(x).

Note that Proposition 7 is a corollary of a theorem of D. P. Geller and

A. J. W. Hilton [8].

COROLLARY 6. Let G

= (

Vi(G), V 2(G), E(G)) be a bipartite graph. If do(x)? dG(Y) for each edge (x, y) with x E V1 ( G), y E V 2 ( G), then

W1 (G)

=

A(G).

COROLLARY 7. Let G=(V1(G), V2(G),E(G)) be a bipartite graph with

ACKNOWLEDGMENT

We appreciate all the suggestions of the referees. In particular, we would like to mention that Theorem 2 has been improved by one of the referees.

REFERENCES

I. A. S. AsRATIAN, Construction of an edge coloring of a special form in a bipartite graph and its applications in scheduling problems, Vestnik Moscow Uniti. Ser. Comput. Math. Cybern. l ( 1978 ), 74-81. [Russian]

2. A. S. AsRATIAN, "Investigation of Some Mathematical Model of Scheduling Theory," Doctoral dissertation, Moscow University, 1980.

3. A. S. AsRATIAN AND R. R. KAMALIAN, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987), 25-34. [Russian]

4. A. S. AsRATIAN, A. V. KosroCHKA, AND A. N. MIRUMIAN, A criterion for the unique colorability of edges of bipartite multigraphs, Discrete Anal. 45 (1987), 3-20. [Russian]

5. A. S. ASRATIAN AND A. N. MrRUMIAN, Transformations of edge colorings of a bipartite multigraph and their applications (Russian), Dok/. Acad. Nauk SSSR 316, No. I (1991 ), 11-12 [Russian] (Engl. transl. Sov. Math. Dok/. 43, No. I (1991 ), 1-3).

6. C. BERGE, "Graphs and Hypergraphs," North-Holland, Amsterdam, 1973.

7. Y. CARO AND J. ScHONHEIM, Generalized I-factorization of trees, Discrete Math. 33

(1981), 319-321.

8. D. P. GELLER AND A. J. W. HILTON, How to color the lines of a bigraph, Networks 4 ( 1974 ), 281-282.

9. F. HARARY, "Graph Theory," Addison-Wesley, Reading, MA, 1969.

10. I. HoLYER, The A·.'3l'-completeness of edge-coloring, SIAM J. Comput. 10, No. 4 (1981), 718-720.

11. R. R. KAMALIAN, "Interval Edge-Colorings of Graphs," Doctoral dissertation, Novo-sibirsk, 1990.

12. R. R. KAMALIAN, Interval Colorings of complete bipartite graphs and trees, [Russian]; preprint of the Computing Centre of the Acad. of Sci. of the Republic of Armenia, Yerevan, 1989.

13. R. M. KARP, Reducibility among Combinatorial Problems, in "Complexity of Computer Computations" (R. E. Miller and J. W. Thatcher, Eds.), pp. 85-103, New York, Plenum, 1972.

14. M. KUBALE, Interval Vertex-colorings of a graph with forbidden colors, Discrete Math. 74

( 1989), 125-136.

15. D. LEVEN AND Z. GAUL, .. !-"&'-completeness of finding the chromatic index of regular graphs, J. Algorithms 4 ( 1983 ), 35-44.

16. S. V. SEVASTIANOV, On completeness of interval coloring of edges of a bipartite graph, manuscript, 1990, Novosibirsk.

17. V. G. VIZING, The chromatic index of a multigraph, Kibernetika 3 (1965 ), 29-39. 18. D. DE WERRA, Investigation on an edge coloring problem, Discrete Math. l (1971 ),

167-179.

19. D. DE WERRA AND A. HERTZ, Consecutive colorings of graphs, Z. Oper. Res. 32 (1988), 1-8.