Simple reduction of - Semantic Scholar
Simple Reduction of f-Colorings to Edge-Colorings Xiao Zhou 1 and Takao Nishizekf2 1 Education Center for Information Porcessing 2 Graduate School of Information Sciences Tohoku University, Sendal 980-77, Japan
A b s t r a c t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordinary edge-coloring problem, that is, we show that, given a graph G = (V,E) and a function f : V -4 N, one can directly construct in polynomial-time a new simple graph whose edge-coloring using a minimum number of colors immediately induces an f-coloring of G using a minimum number of colors. As by-products, we give a necessary and sufficient condition for a graph to have an ffactorization, and show that the edge-coloring problem for multigraphs can be easily reduced to edge-coloring problems for simple graphs.
1
Introduction
An edge-coloring of a graph G = (V, E) is to color all the edges of G so t h a t no two adjacent edges are colored with the same color. The minimum number of colors needed for an edge-coloring is called the chromatic index of G and denoted by xr(G). T h r o u g h o u t the paper the maximum degree of a graph G is denoted by A(G) or simply by A. KSnig showed that x'(G) = A if G is bipartite [3, 6]. Vizing showed t h a t xI(G) = A or A + 1 if G is a simple graph, t h a t is, G has no multiple edges or self-loops [3, 8]. The edge-coloring problem is to find an edge-coloring of G using x~(G) colors. Let f : V -4 N be a function which assigns a natural number f(v) E N to each vertex v E V. Then an f-coloring of G is to color all the edges of G so that, for each vertex v E V, at most f(v) edges incident to v are colored with the same color. Thus an f-coloring of G is a partition of the edge set of G into subsets, each inducing a spanning subgraph whose vertex-degrees are bounded by f . Figure 1 (a) illustrates an f-coloring of a graph with three colors, which are indicated by solid, bold and dashed lines. An ordinary edge-coloring is a special case of an f-coloring such that f(v) = 1 for every vertex v E V. The minimum number of colors needed for an f-coloring is called the f-chromatic index of G and denoted by x~(G). The f-coloring problem is to find an f-coloring of G using X~ (G) colors for a given graph G. Let Af(G) = m a x ~ e y [d(v)/f(v)] where d(v) is the degree of vertex v. It is known t h a t x)(G) = Af or A f + 1 for any simple graph G [4].
224
!
f (v3)=l
f(v2)=2
v4 m~._ f(~)=2
_~.~
f(h) =3
(b) 6/
(a) G
Fig. 1. Transformation from G to G~.
Since the ordinary edge-coloring problem is NP-complete [5], the f-coloring problem is also NP-complete. Therefore the theory of NP-completeness immediately implies that there exists a polynomiM-time reduction of the f-coloring problem to the ordinary edge-coloring problem plausibly through another NPcomplete problem, say 3-SAT. However, no simple direct reduction has been known so far.
1@
•1
I
(a) G
] (b)
VI
Fig. 2. Trivial reduction.
We have given the following trivial reduction [9]. For each vertex v E V of a graph G, replace v with f(v) copies vl, v2,..., v:(.), and attach to the copies the d(v) edges which were incident to v; attach [d(v)/f(v)] or [d(v)/f(v)J edges to each copy vi, 1 < i < f(v). Let G / b e the resulting graph. Figure 2 illustrates G and G/. Clearly A(G]) = AI(G) = max~e V [d(v)/f(v)]. Since an edge-coloring of Gy immediately induces an f-coloring of G using the same number of colors, we have
x (G) < x'(a:).
(1)
If G is simple then G / i s also simple, and ifG is bipartite then G / i s also bipartite. Therefore, the results of Vizing and Khnig together with the reduction above immediately imply that x~(G) = A/(G) or AI(G) + 1 if G is simple and that x~(G) = A/(G) if G is bipartite. Thus the reduction is trivial but very useful.
225
However, Eq.(1) does not always hold in equality. For example, x~(G) = 2 for a graph G in Figure 2(a) as indicated by solid and dashed lines, but x'(Gf) = 3 for a graph Gf in Figure 2(b). In this paper we first give a very simple reduction of the f-coloring problem to the ordinary edge-coloring problem. That is, we show that, given a simple graph G together with a function f : V --r N, one can directly construct in polynomialtime a new simple graph G*f such that X'f (G) -- x'(G*f). We construct G~ from G by inserting an appropriate bipartite graph Pv for each vertex v E V as illustrated in Figure 1. It should be noted that the theory of NP-completeness does not imply the existence of such a single graph G~. We then show that the f-coloring problem for a multigraph can be directly reduced in polynomial time to the edge-coloring problem for several simple graphs. Thus we show that the f coloring problem is not more intractable than the ordinary edge-coloring problem although the former looks to be more difficult than the latter. Furthermore the simple reduction above immediately yields a necessary and sufficient condition for a graph G to have an f-factorization, that is, a partition of G to spanning subgraphs in each of which d(v) = f(v) for every vertex v. Finding such a condition has been an open problem in graph theory [1]. The reduction above also implies that the edge-coloring problem for multigraphs can be easily reduced to the edge-coloring problem for simple graphs. 2
Preliminaries
In this section we give some definitions. Let G = (V, E) denote a graph with vertex set V and edge set E. We often denote by V(G) and E(G) the vertex set and the edge set of G, respectively. We assume that G has no selfloops but may have multiple edges, that is, G is a so-called multigraph. If G has no multiple edges, then G is called a simple graph. An edge joining vertices u and v is denoted by (u, v). The degree of vertex v E V(G) is denoted by d(v, G) or simply by d(v). The maximum degree of G is denoted by A(G), or simply by A. Clearly x'(G) > A(G). A graph G = (V, E) is bipartite if V is bipartitioned into two subsets U and W so that v E U and w E W for every edge (v, w) E E. By Konig's theorem x'(G) = A(G) if G is bipartite [3, 6]. Let f be a function which assigns a natural number f(v) to each vertex v E V. One may assume without loss of generality that f(v) 3. In this section we give a sophisticated reduction for which Eq.(1) always holds in equality. That is, as the main result of this paper, we give the following theorem. T h e o r e m 2. Given any simple graph G = (V, E) and function f such that Af(C) >_ 3, one can directly construct a simple graph G*/ = (V/,E~) such that x~(G) = x'(G~) and IE;I is polynomial in IEI.
It should be noted that, given an edge-coloring of G~ with x'(G*f) colors, one can find in polynomial time an f-coloring of G using x~(G) colors. Thus the fcoloring problem for a simple graph can be reduced to the ordinary edge-coloring problem for a simple graph in polynomial-time.
Iz1 ~
!ell~ ~i2
UOt~ ttX~ ' ~
e~ ~ w1 eo2~ W2 eoff,~ Wot
Fig. 3. (a, fl)-permutation graph P~a. We use the following graph P~# called an (cz,/3)-permutation graph as a building-block to construct G~ from G. See Figure 3. For positive integers a and /3, let P~# be a bipartite simple graph such that 9 there are a input vertices U = { u l , u 2 , . . . , u s } and a input edges Ei = {e~l, ei2," " , ei~} incident to input vertices; 9 there are a output vertices W = {Wl, w2,...,w~} and c~ output edges Eo = {eol, eo2,. 9 eoa} incident to output vertices;
d(v, P ~ ) =
1 ifv~UtJW; /3 otherwise.
Thus A ( P ~ ) =/3. Let C = {Cl,O2,"" ,c~} be any set of/3 colors. We call P~Z an (a, ~3)-permutation graph if (i) for any edge-coloring of P~a with /3 colors, the sequence of colors of output edges is a permutation of that of input edges; and
227
(ii) for any sequence of colors Ci = {c~1,ci2,... ,ci~}, cij ~ C, and for any permutation Co = {Col,Co2," ", cos} of Ci, there is an edge-coloring of P~Z with the fl colors such that input edge e~j is colored cij and output edge eod is colored cod for each j, 1 < j < a. The following lemma holds on P~Z. The construction of P ~ is similar as that of a well-known Clos permutation network.
For any a > 1 and fl > 3 there is an (a,fl)-permutation graph P~f~ such that IE(P,~f~)] = O(afl 2 [log~(a + 1)]).
Lemlna3.
For any integer fl > 3, we construct G~Z from G and copies of P~Z as follows (see Figure 1): (a) replace each vertex v E V by a copy P~ of a (d(v), fl)-permutation graph Pa(.)Z; (b) merge the d(v) output vertices wi, w2,..., Wd@) of P~, v E V, into f(v) vertices v l , v 2 , . . . , v f ( , ) so that d(vj,G*fz) = Ld(v,G)/f(v)J or rd(v,G)/f(v)l for every j, 1 G j < f(v); (c) for each edge e = (v, v') E E, identify, as a single edge, an input edge of P. and an input edge of P~, which are surrogates of e. Clearly
fl
= { af(a)
fl > af(G); otherwise.
Furthermore G ~ is a simple graph even if G has multiple edges. Figure l(b) illustrates G}~ for the graph G in Figure l(a). We have the following theorem on G ~ . T h e o r e m 4. Let G = (V, E) be any multigraph and let fl be any integer with fl > 3. Then x~(G)
A b s t r a c t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordinary edge-coloring problem, that is, we show that, given a graph G = (V,E) and a function f : V -4 N, one can directly construct in polynomial-time a new simple graph whose edge-coloring using a minimum number of colors immediately induces an f-coloring of G using a minimum number of colors. As by-products, we give a necessary and sufficient condition for a graph to have an ffactorization, and show that the edge-coloring problem for multigraphs can be easily reduced to edge-coloring problems for simple graphs.
1
Introduction
An edge-coloring of a graph G = (V, E) is to color all the edges of G so t h a t no two adjacent edges are colored with the same color. The minimum number of colors needed for an edge-coloring is called the chromatic index of G and denoted by xr(G). T h r o u g h o u t the paper the maximum degree of a graph G is denoted by A(G) or simply by A. KSnig showed that x'(G) = A if G is bipartite [3, 6]. Vizing showed t h a t xI(G) = A or A + 1 if G is a simple graph, t h a t is, G has no multiple edges or self-loops [3, 8]. The edge-coloring problem is to find an edge-coloring of G using x~(G) colors. Let f : V -4 N be a function which assigns a natural number f(v) E N to each vertex v E V. Then an f-coloring of G is to color all the edges of G so that, for each vertex v E V, at most f(v) edges incident to v are colored with the same color. Thus an f-coloring of G is a partition of the edge set of G into subsets, each inducing a spanning subgraph whose vertex-degrees are bounded by f . Figure 1 (a) illustrates an f-coloring of a graph with three colors, which are indicated by solid, bold and dashed lines. An ordinary edge-coloring is a special case of an f-coloring such that f(v) = 1 for every vertex v E V. The minimum number of colors needed for an f-coloring is called the f-chromatic index of G and denoted by x~(G). The f-coloring problem is to find an f-coloring of G using X~ (G) colors for a given graph G. Let Af(G) = m a x ~ e y [d(v)/f(v)] where d(v) is the degree of vertex v. It is known t h a t x)(G) = Af or A f + 1 for any simple graph G [4].
224
!
f (v3)=l
f(v2)=2
v4 m~._ f(~)=2
_~.~
f(h) =3
(b) 6/
(a) G
Fig. 1. Transformation from G to G~.
Since the ordinary edge-coloring problem is NP-complete [5], the f-coloring problem is also NP-complete. Therefore the theory of NP-completeness immediately implies that there exists a polynomiM-time reduction of the f-coloring problem to the ordinary edge-coloring problem plausibly through another NPcomplete problem, say 3-SAT. However, no simple direct reduction has been known so far.
1@
•1
I
(a) G
] (b)
VI
Fig. 2. Trivial reduction.
We have given the following trivial reduction [9]. For each vertex v E V of a graph G, replace v with f(v) copies vl, v2,..., v:(.), and attach to the copies the d(v) edges which were incident to v; attach [d(v)/f(v)] or [d(v)/f(v)J edges to each copy vi, 1 < i < f(v). Let G / b e the resulting graph. Figure 2 illustrates G and G/. Clearly A(G]) = AI(G) = max~e V [d(v)/f(v)]. Since an edge-coloring of Gy immediately induces an f-coloring of G using the same number of colors, we have
x (G) < x'(a:).
(1)
If G is simple then G / i s also simple, and ifG is bipartite then G / i s also bipartite. Therefore, the results of Vizing and Khnig together with the reduction above immediately imply that x~(G) = A/(G) or AI(G) + 1 if G is simple and that x~(G) = A/(G) if G is bipartite. Thus the reduction is trivial but very useful.
225
However, Eq.(1) does not always hold in equality. For example, x~(G) = 2 for a graph G in Figure 2(a) as indicated by solid and dashed lines, but x'(Gf) = 3 for a graph Gf in Figure 2(b). In this paper we first give a very simple reduction of the f-coloring problem to the ordinary edge-coloring problem. That is, we show that, given a simple graph G together with a function f : V --r N, one can directly construct in polynomialtime a new simple graph G*f such that X'f (G) -- x'(G*f). We construct G~ from G by inserting an appropriate bipartite graph Pv for each vertex v E V as illustrated in Figure 1. It should be noted that the theory of NP-completeness does not imply the existence of such a single graph G~. We then show that the f-coloring problem for a multigraph can be directly reduced in polynomial time to the edge-coloring problem for several simple graphs. Thus we show that the f coloring problem is not more intractable than the ordinary edge-coloring problem although the former looks to be more difficult than the latter. Furthermore the simple reduction above immediately yields a necessary and sufficient condition for a graph G to have an f-factorization, that is, a partition of G to spanning subgraphs in each of which d(v) = f(v) for every vertex v. Finding such a condition has been an open problem in graph theory [1]. The reduction above also implies that the edge-coloring problem for multigraphs can be easily reduced to the edge-coloring problem for simple graphs. 2
Preliminaries
In this section we give some definitions. Let G = (V, E) denote a graph with vertex set V and edge set E. We often denote by V(G) and E(G) the vertex set and the edge set of G, respectively. We assume that G has no selfloops but may have multiple edges, that is, G is a so-called multigraph. If G has no multiple edges, then G is called a simple graph. An edge joining vertices u and v is denoted by (u, v). The degree of vertex v E V(G) is denoted by d(v, G) or simply by d(v). The maximum degree of G is denoted by A(G), or simply by A. Clearly x'(G) > A(G). A graph G = (V, E) is bipartite if V is bipartitioned into two subsets U and W so that v E U and w E W for every edge (v, w) E E. By Konig's theorem x'(G) = A(G) if G is bipartite [3, 6]. Let f be a function which assigns a natural number f(v) to each vertex v E V. One may assume without loss of generality that f(v) 3. In this section we give a sophisticated reduction for which Eq.(1) always holds in equality. That is, as the main result of this paper, we give the following theorem. T h e o r e m 2. Given any simple graph G = (V, E) and function f such that Af(C) >_ 3, one can directly construct a simple graph G*/ = (V/,E~) such that x~(G) = x'(G~) and IE;I is polynomial in IEI.
It should be noted that, given an edge-coloring of G~ with x'(G*f) colors, one can find in polynomial time an f-coloring of G using x~(G) colors. Thus the fcoloring problem for a simple graph can be reduced to the ordinary edge-coloring problem for a simple graph in polynomial-time.
Iz1 ~
!ell~ ~i2
UOt~ ttX~ ' ~
e~ ~ w1 eo2~ W2 eoff,~ Wot
Fig. 3. (a, fl)-permutation graph P~a. We use the following graph P~# called an (cz,/3)-permutation graph as a building-block to construct G~ from G. See Figure 3. For positive integers a and /3, let P~# be a bipartite simple graph such that 9 there are a input vertices U = { u l , u 2 , . . . , u s } and a input edges Ei = {e~l, ei2," " , ei~} incident to input vertices; 9 there are a output vertices W = {Wl, w2,...,w~} and c~ output edges Eo = {eol, eo2,. 9 eoa} incident to output vertices;
d(v, P ~ ) =
1 ifv~UtJW; /3 otherwise.
Thus A ( P ~ ) =/3. Let C = {Cl,O2,"" ,c~} be any set of/3 colors. We call P~Z an (a, ~3)-permutation graph if (i) for any edge-coloring of P~a with /3 colors, the sequence of colors of output edges is a permutation of that of input edges; and
227
(ii) for any sequence of colors Ci = {c~1,ci2,... ,ci~}, cij ~ C, and for any permutation Co = {Col,Co2," ", cos} of Ci, there is an edge-coloring of P~Z with the fl colors such that input edge e~j is colored cij and output edge eod is colored cod for each j, 1 < j < a. The following lemma holds on P~Z. The construction of P ~ is similar as that of a well-known Clos permutation network.
For any a > 1 and fl > 3 there is an (a,fl)-permutation graph P~f~ such that IE(P,~f~)] = O(afl 2 [log~(a + 1)]).
Lemlna3.
For any integer fl > 3, we construct G~Z from G and copies of P~Z as follows (see Figure 1): (a) replace each vertex v E V by a copy P~ of a (d(v), fl)-permutation graph Pa(.)Z; (b) merge the d(v) output vertices wi, w2,..., Wd@) of P~, v E V, into f(v) vertices v l , v 2 , . . . , v f ( , ) so that d(vj,G*fz) = Ld(v,G)/f(v)J or rd(v,G)/f(v)l for every j, 1 G j < f(v); (c) for each edge e = (v, v') E E, identify, as a single edge, an input edge of P. and an input edge of P~, which are surrogates of e. Clearly
fl
= { af(a)
fl > af(G); otherwise.
Furthermore G ~ is a simple graph even if G has multiple edges. Figure l(b) illustrates G}~ for the graph G in Figure l(a). We have the following theorem on G ~ . T h e o r e m 4. Let G = (V, E) be any multigraph and let fl be any integer with fl > 3. Then x~(G)
