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Two Conjectures on Edge-Colouring
Discrete Mathematics 74 (1989) 61-64 North-Holland
61
TWO CONJECTURES ON EDGE-COLOURING A.J.W. HILTON Dept. of Mathematics, University of Reading, Whiteknighrs, P. 0 . Box 220, Reading, RG62AX, U.K. Chetwynd and Hilton have elsewhere posed two conjectures, one a general statement on edge-colouring simple graphs C with A ( G )> f IV(G)l. and a second to the effect that a regular simple graph G with d ( G ) 3 IV(C)l is 1-factorizable. We set out the evidence for both these conjectures and show that the first implies the second.
1. Introduction We are concerned here with simple graphs, that is finite graphs without loops or multiple edges. An edge-colouring of a graph G is a map 4: E ( G ) + %',where %' is a set of colours, such that no two vertices with the same colour have a common vertex. The chromatic index x ' ( G ) is the least value of l%'l for which an edge-colouring exists. A well-known theorem of Vizing [17] states that A ( G ) s x ' ( G )s A ( C ) + 1, where A ( G ) denotes the maximum degree of G. If ~ ' ( ( 3= ) A ( G ) , then G is said to be Class 1, and otherwise G is Class 2. The question of deciding whether or not a graph is Class 1 was shown by Holyer [14] to be NP-complete. However, for certain types of graph, the problem of classifying Class 2 graphs seems to be tractable. If G satisfies the inequality
then G is overfull. Clearly if G is overfull, then IV(G)l is odd. An overfull graph has to be Class 2, since no colour class of G can have more than 1; JV(G)] edges. In [6], Chetwynd and Hilton made the following conjecture (now slightly modified).
Conjecture 1. Let G be a simple graph with A(G) > f IV(G)l. Then G is Class 2 if and only if G contains an overfull subgraph H with A ( H ) = A(G). The graph C obtained from Petersen's graph by removing one vertex is Class 2, but contains no subgraph H with A ( H ) = A ( G ) ; this shows that the figure 3 in Conjecture 1 cannot be lowered. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)
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A.J.W . Hilton
At the time of writing, Conjecture 1 has been proved in a number of cases. Plantholt ( [ 1 5 , 161) and Chetwynd and Hilton ([3-61) have between them established the following.
Theorem 1. Conjecture 1 is true
if A(G) 3 I V(C)l- 3.
In [ l ] , Chetwynd and Hilton posed the following conjecture about regular graphs of even order. First note that a Class 1 regular graph is often called l-factorizable, as it is the union of edge-disjoint l-factors. Also note that a regular graph of odd order is overfull, and so is Class 2 . If a graph G is regular, let d ( G ) denote its degree.
Conjecture 2. Let C be a regular simple graph of even order satisfying d ( G ) 3 1 IV(C)l. Then C is l-factorizable. This conjecture seems to have been known however long before being posed by Chetwynd and myself. When I told Dirac of it, he said it was “going around” in the early 1950s. The figure 12 IV(C)l in the conjecture cannot be lowered, as is shown by the example of a graph C consisting of two K,’s, when n is odd. Chetwynd and Hilton ( [ 1 , 7 , 8 ] )have proved this conjecture in a number of special cases.
Theorem 2. Conjecture 2 is true if either d ( G )3 4
( f i - 1 ) IV(G)l
or d ( G ) 3 IV(C)l- 4. The object of this note is to prove the following theorem.
Theorem 3. If Conjecture 1 is true, then Conjecture 2 is true. 2. Proof of Theorem 3 Let C be a regular graph with IV(C)l = 2n and d ( C ) a a . Suppose that Conjecture 1 is true and that G is Class 2. Let H be an overfull subgraph of G with A ( H ) = d ( G ) . Since H is overfull, it follows that IV(H)I is odd, so H # G. Let def(H) =
2
tJtV(H)
( d ( G )- d H ( V ) ) .
Two conjectures on edge-colouring
63
It is shown in [2] that, if H is overfull, then def(H) S A ( H ) - 2 = d ( C ) - 2. It follows that G has an edge-cut S with )SI G d ( G )- 2 such that G\S = H U J , where V ( H )f Vl ( J )= 9. Since A ( H ) = d ( G ) > n , it follows that H has at least n 1 vertices. Consequently J has at most n - 1 vertices. Thus d ( G ) 1> IV(J)(.Since C is regular, the number of edges joining vertices of J to vertices of H is at least ( d ( G )- IV(J)I 1) IV(J)I. For fixed d ( G ) , ( d ( C )- IV(J)(+ 1) IV(J)I is a quadratic in IV(J)(.In the range 1 s IV(J)(S n - 1, it has two minima, one at each end point, with values d ( G ) and ( d ( G ) - n + 2 ) ( n - 1). But d ( G ) > IS[, and ( d ( C )- n 2)(n - 1) 2 2n - 2 3 d ( G ) - 1> IS(, contradicting the definition of S. Thus C has no overfull subgraph H , and so, by Conjecture 1, is Class 1, or in other words is 1-factorizable. Thus Conjecture 2 is true. This proves Theorem 3.
+
+
+
+
3. A final remark Conjecture 1 has many other implications. Some of these are discussed in [ll-131 by Hilton and Johnson. A survey of the main implications is given in [lo]. See also [9].
Note added in proof A.G. Chetwynd and I have recently proved Conjecture 1 in the case when A G ~ = a ( f i - l)(lV(C)l + 1)+ 1 and IE(C)l= A(G)Li IV(G)ll. See [18].
References [l] A.G. Chetwynd and A.J.W. Hilton, Regular graphs of high degree are 1-factorizable, Proc. London Math. SOC.(3), 50 (1985) 193-206. [Z] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small, J. Combinatorial Theory (B) to appear. [3] A.G. Chetwynd and A.J.W. Hilton, Partial edge-colourings of complete graphs or of graphs which are nearly complete, Graph Theory and Combinatorics. Vol. in honour of P. Erdos’ 70th birthday (1984) 81-98. [4] A.G. Chetwynd and A.J.W. Hilton, The chromatic index of graphs of even order with many edges, J. Graph Theory 8 (1984) 463-470. [5] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with maximum degree at least IVI - 3. Proceedings of the conference held in Denmark in memory of G . A . Dirac, to appear. [6] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with three vertices of maximum degree, Math. Proc. Camb. Phil. SOC.100 (1986) 303-317.
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A.J. W. Hilton
[71 A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of regular graphs of degree 4 and their complements, Discrete Applied Math. 16 (1987) 125-134. 181 . . A.G. Chetwynd and A.J.W. Hilton, 1-factorizing regular graphs of high degree -an improved bound, submitted. [9] A.G. Chetwynd and A.J.W. Hilton, A A-subgraph condition for a graph to be Class 1, J. Combinatorial Theory (B) to appear. [lo] A.J.W. Hilton, Recent progress on edge-colouring graphs, Discrete Math. 64 (1987) 303-307. [ l l ] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic number, Math. Proc. Camb. Phil. SOC., 102 (1987) 211-221. [12] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic class, submitted. [13] A.J. Hilton and P.D. Johnson, Reverse class critical multigraphs, Discrete Math. 69 (1988) 309-311. [14] I.J. Holyer, The NP-completeness of edge-colourings, SIAM J. Computing 10 (1980) 718-720. [15] M. Plantholt, The chromatic index of graphs with a spanning star, J. Graph Theory 5 (1981) 5-13. [16] M. Plantholt, On the chromatic index of graphs with large maximum degree, Discrete Math. 47 (1983) 91-96. [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz. 3 (1964) 25-30. [l8] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with r vertices of maximum degree, submitted.
61
TWO CONJECTURES ON EDGE-COLOURING A.J.W. HILTON Dept. of Mathematics, University of Reading, Whiteknighrs, P. 0 . Box 220, Reading, RG62AX, U.K. Chetwynd and Hilton have elsewhere posed two conjectures, one a general statement on edge-colouring simple graphs C with A ( G )> f IV(G)l. and a second to the effect that a regular simple graph G with d ( G ) 3 IV(C)l is 1-factorizable. We set out the evidence for both these conjectures and show that the first implies the second.
1. Introduction We are concerned here with simple graphs, that is finite graphs without loops or multiple edges. An edge-colouring of a graph G is a map 4: E ( G ) + %',where %' is a set of colours, such that no two vertices with the same colour have a common vertex. The chromatic index x ' ( G ) is the least value of l%'l for which an edge-colouring exists. A well-known theorem of Vizing [17] states that A ( G ) s x ' ( G )s A ( C ) + 1, where A ( G ) denotes the maximum degree of G. If ~ ' ( ( 3= ) A ( G ) , then G is said to be Class 1, and otherwise G is Class 2. The question of deciding whether or not a graph is Class 1 was shown by Holyer [14] to be NP-complete. However, for certain types of graph, the problem of classifying Class 2 graphs seems to be tractable. If G satisfies the inequality
then G is overfull. Clearly if G is overfull, then IV(G)l is odd. An overfull graph has to be Class 2, since no colour class of G can have more than 1; JV(G)] edges. In [6], Chetwynd and Hilton made the following conjecture (now slightly modified).
Conjecture 1. Let G be a simple graph with A(G) > f IV(G)l. Then G is Class 2 if and only if G contains an overfull subgraph H with A ( H ) = A(G). The graph C obtained from Petersen's graph by removing one vertex is Class 2, but contains no subgraph H with A ( H ) = A ( G ) ; this shows that the figure 3 in Conjecture 1 cannot be lowered. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)
62
A.J.W . Hilton
At the time of writing, Conjecture 1 has been proved in a number of cases. Plantholt ( [ 1 5 , 161) and Chetwynd and Hilton ([3-61) have between them established the following.
Theorem 1. Conjecture 1 is true
if A(G) 3 I V(C)l- 3.
In [ l ] , Chetwynd and Hilton posed the following conjecture about regular graphs of even order. First note that a Class 1 regular graph is often called l-factorizable, as it is the union of edge-disjoint l-factors. Also note that a regular graph of odd order is overfull, and so is Class 2 . If a graph G is regular, let d ( G ) denote its degree.
Conjecture 2. Let C be a regular simple graph of even order satisfying d ( G ) 3 1 IV(C)l. Then C is l-factorizable. This conjecture seems to have been known however long before being posed by Chetwynd and myself. When I told Dirac of it, he said it was “going around” in the early 1950s. The figure 12 IV(C)l in the conjecture cannot be lowered, as is shown by the example of a graph C consisting of two K,’s, when n is odd. Chetwynd and Hilton ( [ 1 , 7 , 8 ] )have proved this conjecture in a number of special cases.
Theorem 2. Conjecture 2 is true if either d ( G )3 4
( f i - 1 ) IV(G)l
or d ( G ) 3 IV(C)l- 4. The object of this note is to prove the following theorem.
Theorem 3. If Conjecture 1 is true, then Conjecture 2 is true. 2. Proof of Theorem 3 Let C be a regular graph with IV(C)l = 2n and d ( C ) a a . Suppose that Conjecture 1 is true and that G is Class 2. Let H be an overfull subgraph of G with A ( H ) = d ( G ) . Since H is overfull, it follows that IV(H)I is odd, so H # G. Let def(H) =
2
tJtV(H)
( d ( G )- d H ( V ) ) .
Two conjectures on edge-colouring
63
It is shown in [2] that, if H is overfull, then def(H) S A ( H ) - 2 = d ( C ) - 2. It follows that G has an edge-cut S with )SI G d ( G )- 2 such that G\S = H U J , where V ( H )f Vl ( J )= 9. Since A ( H ) = d ( G ) > n , it follows that H has at least n 1 vertices. Consequently J has at most n - 1 vertices. Thus d ( G ) 1> IV(J)(.Since C is regular, the number of edges joining vertices of J to vertices of H is at least ( d ( G )- IV(J)I 1) IV(J)I. For fixed d ( G ) , ( d ( C )- IV(J)(+ 1) IV(J)I is a quadratic in IV(J)(.In the range 1 s IV(J)(S n - 1, it has two minima, one at each end point, with values d ( G ) and ( d ( G ) - n + 2 ) ( n - 1). But d ( G ) > IS[, and ( d ( C )- n 2)(n - 1) 2 2n - 2 3 d ( G ) - 1> IS(, contradicting the definition of S. Thus C has no overfull subgraph H , and so, by Conjecture 1, is Class 1, or in other words is 1-factorizable. Thus Conjecture 2 is true. This proves Theorem 3.
+
+
+
+
3. A final remark Conjecture 1 has many other implications. Some of these are discussed in [ll-131 by Hilton and Johnson. A survey of the main implications is given in [lo]. See also [9].
Note added in proof A.G. Chetwynd and I have recently proved Conjecture 1 in the case when A G ~ = a ( f i - l)(lV(C)l + 1)+ 1 and IE(C)l= A(G)Li IV(G)ll. See [18].
References [l] A.G. Chetwynd and A.J.W. Hilton, Regular graphs of high degree are 1-factorizable, Proc. London Math. SOC.(3), 50 (1985) 193-206. [Z] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small, J. Combinatorial Theory (B) to appear. [3] A.G. Chetwynd and A.J.W. Hilton, Partial edge-colourings of complete graphs or of graphs which are nearly complete, Graph Theory and Combinatorics. Vol. in honour of P. Erdos’ 70th birthday (1984) 81-98. [4] A.G. Chetwynd and A.J.W. Hilton, The chromatic index of graphs of even order with many edges, J. Graph Theory 8 (1984) 463-470. [5] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with maximum degree at least IVI - 3. Proceedings of the conference held in Denmark in memory of G . A . Dirac, to appear. [6] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with three vertices of maximum degree, Math. Proc. Camb. Phil. SOC.100 (1986) 303-317.
64
A.J. W. Hilton
[71 A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of regular graphs of degree 4 and their complements, Discrete Applied Math. 16 (1987) 125-134. 181 . . A.G. Chetwynd and A.J.W. Hilton, 1-factorizing regular graphs of high degree -an improved bound, submitted. [9] A.G. Chetwynd and A.J.W. Hilton, A A-subgraph condition for a graph to be Class 1, J. Combinatorial Theory (B) to appear. [lo] A.J.W. Hilton, Recent progress on edge-colouring graphs, Discrete Math. 64 (1987) 303-307. [ l l ] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic number, Math. Proc. Camb. Phil. SOC., 102 (1987) 211-221. [12] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic class, submitted. [13] A.J. Hilton and P.D. Johnson, Reverse class critical multigraphs, Discrete Math. 69 (1988) 309-311. [14] I.J. Holyer, The NP-completeness of edge-colourings, SIAM J. Computing 10 (1980) 718-720. [15] M. Plantholt, The chromatic index of graphs with a spanning star, J. Graph Theory 5 (1981) 5-13. [16] M. Plantholt, On the chromatic index of graphs with large maximum degree, Discrete Math. 47 (1983) 91-96. [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz. 3 (1964) 25-30. [l8] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with r vertices of maximum degree, submitted.