Asymptotically good colourings of graphs and multigraphs∗ P. Mark Kayll† Department of Mathematical Sciences, The University of Montana Missoula, MT 59812-1032, USA [email protected]

Abstract A multigraph invariant β admitting formulation as the solution to an integer program is asymptotically good, or a.g., in case β/β ∗ → 1 as β ∗ → ∞ (β ∗ is the solution of the linear relaxation of the IP defining β). Recent developments establishing the a.g. behaviour of three colouring parameters — the chromatic index, the list-chromatic index and the total chromatic number — are surveyed. New here is the addition of the last invariant to the list; two short proofs of this observation are presented. One is based on a theorem of Kahn (J. Combin. Theory Ser. A 73 (1996), 1–59) concerning the asymptotics of the list-chromatic index, the other on recent progress of Molloy and Reed in the direction of the BehzadVizing Conjecture. That our main result applies only to graphs invites attention to the more general case of multigraphs.

1

Introduction

This paper has threefold purpose. One aim is to point out (Theorem 1.1) and prove (§4) an asymptotic connection between the total chromatic number χt and its fractional analogue χ∗t for (simple) graphs with large maximum degree. (Definitions are postponed until later in this section.) Recent and far-reaching progress [24, 40] in different areas of colouring theory leads to two independent, satisfyingly simple, proofs of our main result: 1991 AMS Subject Classification: Primary 05C15, 05C70; Secondary 90C5, 90C10. paper is a (substantially) revised version of [34]. † Supported in part by the University Grant Program, The University of Montana.

∗ This

1

(1.1) Theorem For graphs G, χt (G) ∼ χ∗t (G)

as χ∗t (G) → ∞.

(1)

That is, for each ε > 0 there exists D = D(ε) such that every (simple) graph G with χ∗t (G) > D satisfies (1 + ε)−1

0 there exists B = B(ε) such that (1 + ε)−1

0 there exists d = d(γ) such that every graph G with ∆(G) > d satisfies χ′ℓ (G) < (1 + γ)∆(G). Theorem 4.3 is actually a corollary of a vastly more general result of Kahn, who proved the theorem for hypergraphs of bounded edge size that satisfy a natural local sparseness condition. Even in the more general setting, it is true — but not obvious; see [26] — that χ′∗ ∼ ∆, so that [24] asserts the a.g. behaviour of ℓ χ′ℓ for certain hypergraphs, including graphs (cf. Theorem 3.2). For our purposes, i.e. for the first proof of Theorem 1.1, we need only the special case stated here. Our second proof will be more direct, exploiting a recent result of Molloy and Reed [40] in the direction of Conjecture 2.3. The latest in a sequence (surveyed in [22] and recently augmented by [8, 19]) of improvements to upper bounds on χt , their result is (4.4) Theorem There is a constant C such that every graph satisfies χt ≤ ∆ + C. It is a nontrivial undertaking to determine the optimal value of C offered by the proof of Theorem 4.4, but the authors do reveal that it is at least 500. Though this may leave room for improvement towards Conjecture 2.3, this tremendous result gives substantially more than we need for proving Theorem 1.1 (see the remark following the second proof). Two proofs of main result The contrast between the complexity of the proofs of Theorems 3.1, 3.2 and that of the following proofs derives from the power of Theorems 4.3, 4.4: with the foundations laid (in §§1, 4), each yields Theorem 1.1 as a corollary. First proof. In light of (9), it remains only to establish the right-hand inequality in (2) for sufficiently large χ∗t . Given ε > 0, let γ = ε/2, and choose d so large (according to Theorem 4.3) that ∆>d

χ′ℓ < (1 + γ)∆.

implies

8

(10)

Provided χ∗t > D := max{d + 2, 2/γ}, we then have χt ≤ χ′ℓ + 2 < (1 + γ)∆ + 2 < (1 + γ)χ∗t + γχ∗t = (1 + ε)χ∗t (justifying the inequalities, resp., by: Lemma 4.2; (10), Theorem 4.1 and χ∗t > d + 2; (8) and χ∗t > 2/γ), which is what we want. Second proof. We dispense with the “epsilontics” and prove (1) directly. Combining (8), (9) and Theorem 4.4, we have ∆ + 1 ≤ χ∗t ≤ χt ≤ ∆ + C

(11)

(where C is a constant), from which it is obvious that χt ∼ χ∗t (∼ ∆) as ∆ → ∞, and that ∆ → ∞ if and only if χ∗t → ∞. Notice that the second proof would still carry through even if the upper bound for χt in (11) were replaced by ∆ + o(∆) (with o(∆) any function such that o(∆)/∆ → 0 as ∆ → ∞). Such bounds were obtained previously by Hind [17], Chetwynd and H¨ aggkvist [8], and Hind et al. [19].

5

Speculation

In closing, we suggest several avenues for future research. Since our proofs of Theorem 1.1 apply only to graphs, one natural problem is to generalize this result to multigraphs: (5.1) Conjecture For multigraphs, χt is a.g. As numerous colouring invariants beyond those we have considered have been proposed, there is a sense in which Theorems 1.1, 3.1, 3.2 scratch only the surface of possibilities for a.g. multigraph colouring. To illustrate, we introduce the entire chromatic number, χvef , defined, for a plane multigraph G, to be the least t for which G admits a t-colouring of its vertices, edges and faces. One also has χve , χvf , χef , etc., each defined in the natural way; observe that we have denoted χve by χt and χe by χ′ . For references and background on some of these more “exotic” colouring invariants, see, e.g., [7, 22, 50]. Faced with such an abundance of parameters, we are unable to resist proposing several conjectures reflecting the key theme of this paper. (5.2) Conjecture For plane multigraphs, χvef is a.g. By abuse of notation, we condense into one statement two additional conjectures in the spirit of Conjecture 5.1: 9

(5.3) Conjecture For plane multigraphs, χmf is a.g., where m denotes either v or e. Quite possibly Conjectures 5.2, 5.3 are easily handled (or worse, silly). Moreover, as the author is unaware of the computational complexity of χvef , χvf , χef , it is not clear (to him) that they even share the same motivational basis as do Theorems 1.1, 3.1, 3.2. They are offered more for their potential to provide insight into Problem 5.4 than as a reflection of the author’s belief in their truth; indeed, they suggested themselves as this paper was being completed, and no serious attempts have been made to resolve any of our own conjectures. A glaring omission from our list of (multi)graph colouring invariants exhibiting a.g. behaviour is χ, whose determination is of course N P-hard (see [31] or, e.g., [13]). Thus, it is natural to pose the following (5.4) Problem Determine conditions on G to ensure that χ(G) is a.g.

(12)

Notice that two partial solutions to Problem 5.4 have already appeared in this paper. Theorem 3.1 shows that “G is the line graph of a multigraph” suffices for (12), while Theorem 1.1 offers the sufficient condition “G is the total graph of a (simple) graph”. It is intriguing to seek alternate sufficient conditions, or perhaps even a complete characterization of (12). We leave this for the reader to ponder. Acknowledgements The author thanks Jeannette Janssen, Jeff Kahn and Mike Molloy for making available as yet unpublished manuscripts. Thanks also to Karen Seyffarth, Marek Kubale and Glenn Hurlbert for assistance with some of the other references.

References [1] N. Alon, Restricted colorings of graphs, in “Surveys in Combinatorics, Keele, 1993” (K. Walker, Ed.), London Math. Soc. Lecture Note Ser. 187 (1993), 1–33. [2] L.D. Anderson, On edge-colourings of graphs, Math. Scand. 40 (1977), 161–175. [3] M. Behzad, Graphs and Their Chromatic Numbers, Dissertation, Michigan State University, East Lansing, MI, 1965. [4] C. Berge, Hypergraphs: Combinatorics of Finite Sets, NorthHolland, Amsterdam, 1989. 10

´s, Graph Theory: An Introductory Course, Springer[5] B. Bolloba Verlag, New York, 1979. ´s and A.J. Harris, List colorings of graphs, Graphs [6] B. Bolloba Combin. 1 (1985), 115–127. [7] O.V. Borodin, Structural theorem on plane graphs with application to the entire coloring number, J. Graph Theory 23 (1996), 233–239. ¨ggkvist, An improvement of Hind’s up[8] A. Chetwynd and R. Ha per bound on the total chromatic number, Combin. Probab. Comput. 5 (1996), 99–104. ´tal, Linear Programming, W.H. Freeman and Company, [9] V. Chva New York, 1983. [10] M.N. Ellingham and L. Goddyn, List edge colourings of some 1-factorable multigraphs, Combinatorica 16 (1996), 343–352. ˝ s, A. Rubin and H. Taylor, Choosability in graphs, [11] P. Erdo Congr. Numer. 26 (1979), 125–157. [12] F. Galvin, The list chromatic index of a bipartite multigraph, J. Combin. Theory Ser. B 63 (1995), 153–158. [13] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, San Francisco, 1979. [14] M.K. Goldberg, Multigraphs with a chromatic index that is nearly maximal (in Russian), Diskret. Analiz 23 (1973), 3–7. ¨ggkvist and J.C.M. Janssen, On the list-chromatic index [15] R. Ha of bipartite graphs, unpublished manuscript. ¨ggkvist and J.C.M. Janssen, New bounds on the list[16] R. Ha chromatic index of the complete graph, Combin. Probab. Comput., to appear. [17] H. Hind, An upper bound for the total chromatic number, Graphs Combin. 6 (1990), 153–159. [18] H. Hind, Recent developments in total colouring, Discrete Math. 125 (1994), 211–218. [19] H. Hind, M. Molloy and B. Reed, Total colouring with ∆ + poly(log ∆) colours, submitted.

11

[20] H. Hind, M. Molloy and B. Reed, Colouring a graph frugally, submitted. [21] I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981), 718–720. [22] T. Jensen and B. Toft, Graph Coloring Problems, Wiley, New York, 1995. [23] J. Kahn, Recent results on some not-so-recent hypergraph matching and covering problems, in “Extremal Problems for Finite Sets, Visegr´ ad, 1991” (P. Frankl, Z. F¨ uredi, G. Katona and D. Mikl´ os, Eds.), Bolyai Soc. Math. Studies 3 (1994), 305–353. [24] J. Kahn, Asymptotically good list-colorings, J. Combin. Theory Ser. A 73 (1996), 1–59. [25] J. Kahn, A linear programming perspective on the Frankl-R¨ odlPippenger theorem, Random Struct. Alg. 8 (1996), 149–157. [26] J. Kahn, Asymptotics of the chromatic index for multigraphs, J. Combin. Theory Ser. B 68 (1996), 233–254. [27] J. Kahn, Asymptotics of the list-chromatic index for multigraphs, submitted. [28] J. Kahn and P.M. Kayll, Fractional v. integral covers in hypergraphs of bounded edge size, J. Combin. Theory Ser. A, to appear. [29] J. Kahn and P.M. Kayll, On the stochastic independence properties of hard-core distributions, Combinatorica, to appear. [30] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373–395. [31] R.M. Karp, Reducibility among combinatorial problems, in “Complexity of Computer Computations” (R.E. Miller and J.W. Thatcher, Eds.), pp. 85–103, Plenum Press, New York, 1972. [32] P.M. Kayll, Asymptotically Good Covers in Hypergraphs, Dissertation, Rutgers University, New Brunswick, NJ, 1994. [33] P.M. Kayll, “Normal” distributions on matchings in a multigraph: overview with applications, Congr. Numer. 107 (1995), 179–191. [34] P.M. Kayll, Asymptotics of the total chromatic number for simple graphs, Technical Report #96-20, DIMACS, Piscataway, NJ, 1996.

12

[35] P.M. Kayll, Asymptotically good covers in hypergraphs, Diss. Summ. Math. 1 (1996), 9–16. [36] L.G. Khachian, A polynomial algorithm in linear programming (in Russian), Dokl. Acad. Nauk SSSR 244 (1979), 1093–1096 [English translation: Soviet Math. Dokl. 20 (1979), 191–194]. [37] K. Kilakos and B. Reed, Fractionally colouring total graphs, Combinatorica 13 (1993), 435–440. [38] M. Kubale, Some results concerning the complexity of restricted colorings of graphs, Discrete Appl. Math. 36 (1992), 35–46. ´sz and M.D. Plummer, Matching Theory, North-Holland, [39] L. Lova New York, 1986. [40] M. Molloy and B. Reed, A bound on the total chromatic number, Technical Report 304/96, Dept. of Computer Science, University of Toronto, 1996. [41] N. Pippenger and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combin. Theory Ser. A 51 (1989), 24–42. [42] J. Ryan, Fractional total colouring, Discrete Appl. Math. 27 (1990), 287–292. ´nchez-Arroyo, Determining the total colouring number is [43] A. Sa NP-hard, Discrete Math. 78 (1989), 315–319. [44] A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1986. [45] P.D. Seymour, Some unsolved problems on one-factorizations of graphs, in “Graph Theory and Related Topics, Waterloo, 1977” (J.A. Bondy and U.S.R. Murty, Eds.), pp. 367–368, Academic Press, New York, 1979. [46] T. Slivnik, Short proof of Galvin’s theorem on the list-chromatic index of a bipartite multigraph, Combin. Probab. Comput. 5 (1996), 91–94. [47] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz 3 (1964), 25–30. [48] V.G. Vizing, Some unsolved problems in graph theory, Russian Math Surveys 23 (1968), 125–141.

13

[49] V.G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz 29 (1976), 3–10. [50] W. Wang, On the colorings of outerplanar graphs, Discrete Math. 147 (1995), 257–269. [51] H.P. Yap, Total Colourings of Graphs, Lecture Notes in Mathematics 1623, Springer, New York, 1996.

14

Abstract A multigraph invariant β admitting formulation as the solution to an integer program is asymptotically good, or a.g., in case β/β ∗ → 1 as β ∗ → ∞ (β ∗ is the solution of the linear relaxation of the IP defining β). Recent developments establishing the a.g. behaviour of three colouring parameters — the chromatic index, the list-chromatic index and the total chromatic number — are surveyed. New here is the addition of the last invariant to the list; two short proofs of this observation are presented. One is based on a theorem of Kahn (J. Combin. Theory Ser. A 73 (1996), 1–59) concerning the asymptotics of the list-chromatic index, the other on recent progress of Molloy and Reed in the direction of the BehzadVizing Conjecture. That our main result applies only to graphs invites attention to the more general case of multigraphs.

1

Introduction

This paper has threefold purpose. One aim is to point out (Theorem 1.1) and prove (§4) an asymptotic connection between the total chromatic number χt and its fractional analogue χ∗t for (simple) graphs with large maximum degree. (Definitions are postponed until later in this section.) Recent and far-reaching progress [24, 40] in different areas of colouring theory leads to two independent, satisfyingly simple, proofs of our main result: 1991 AMS Subject Classification: Primary 05C15, 05C70; Secondary 90C5, 90C10. paper is a (substantially) revised version of [34]. † Supported in part by the University Grant Program, The University of Montana.

∗ This

1

(1.1) Theorem For graphs G, χt (G) ∼ χ∗t (G)

as χ∗t (G) → ∞.

(1)

That is, for each ε > 0 there exists D = D(ε) such that every (simple) graph G with χ∗t (G) > D satisfies (1 + ε)−1

0 there exists B = B(ε) such that (1 + ε)−1

0 there exists d = d(γ) such that every graph G with ∆(G) > d satisfies χ′ℓ (G) < (1 + γ)∆(G). Theorem 4.3 is actually a corollary of a vastly more general result of Kahn, who proved the theorem for hypergraphs of bounded edge size that satisfy a natural local sparseness condition. Even in the more general setting, it is true — but not obvious; see [26] — that χ′∗ ∼ ∆, so that [24] asserts the a.g. behaviour of ℓ χ′ℓ for certain hypergraphs, including graphs (cf. Theorem 3.2). For our purposes, i.e. for the first proof of Theorem 1.1, we need only the special case stated here. Our second proof will be more direct, exploiting a recent result of Molloy and Reed [40] in the direction of Conjecture 2.3. The latest in a sequence (surveyed in [22] and recently augmented by [8, 19]) of improvements to upper bounds on χt , their result is (4.4) Theorem There is a constant C such that every graph satisfies χt ≤ ∆ + C. It is a nontrivial undertaking to determine the optimal value of C offered by the proof of Theorem 4.4, but the authors do reveal that it is at least 500. Though this may leave room for improvement towards Conjecture 2.3, this tremendous result gives substantially more than we need for proving Theorem 1.1 (see the remark following the second proof). Two proofs of main result The contrast between the complexity of the proofs of Theorems 3.1, 3.2 and that of the following proofs derives from the power of Theorems 4.3, 4.4: with the foundations laid (in §§1, 4), each yields Theorem 1.1 as a corollary. First proof. In light of (9), it remains only to establish the right-hand inequality in (2) for sufficiently large χ∗t . Given ε > 0, let γ = ε/2, and choose d so large (according to Theorem 4.3) that ∆>d

χ′ℓ < (1 + γ)∆.

implies

8

(10)

Provided χ∗t > D := max{d + 2, 2/γ}, we then have χt ≤ χ′ℓ + 2 < (1 + γ)∆ + 2 < (1 + γ)χ∗t + γχ∗t = (1 + ε)χ∗t (justifying the inequalities, resp., by: Lemma 4.2; (10), Theorem 4.1 and χ∗t > d + 2; (8) and χ∗t > 2/γ), which is what we want. Second proof. We dispense with the “epsilontics” and prove (1) directly. Combining (8), (9) and Theorem 4.4, we have ∆ + 1 ≤ χ∗t ≤ χt ≤ ∆ + C

(11)

(where C is a constant), from which it is obvious that χt ∼ χ∗t (∼ ∆) as ∆ → ∞, and that ∆ → ∞ if and only if χ∗t → ∞. Notice that the second proof would still carry through even if the upper bound for χt in (11) were replaced by ∆ + o(∆) (with o(∆) any function such that o(∆)/∆ → 0 as ∆ → ∞). Such bounds were obtained previously by Hind [17], Chetwynd and H¨ aggkvist [8], and Hind et al. [19].

5

Speculation

In closing, we suggest several avenues for future research. Since our proofs of Theorem 1.1 apply only to graphs, one natural problem is to generalize this result to multigraphs: (5.1) Conjecture For multigraphs, χt is a.g. As numerous colouring invariants beyond those we have considered have been proposed, there is a sense in which Theorems 1.1, 3.1, 3.2 scratch only the surface of possibilities for a.g. multigraph colouring. To illustrate, we introduce the entire chromatic number, χvef , defined, for a plane multigraph G, to be the least t for which G admits a t-colouring of its vertices, edges and faces. One also has χve , χvf , χef , etc., each defined in the natural way; observe that we have denoted χve by χt and χe by χ′ . For references and background on some of these more “exotic” colouring invariants, see, e.g., [7, 22, 50]. Faced with such an abundance of parameters, we are unable to resist proposing several conjectures reflecting the key theme of this paper. (5.2) Conjecture For plane multigraphs, χvef is a.g. By abuse of notation, we condense into one statement two additional conjectures in the spirit of Conjecture 5.1: 9

(5.3) Conjecture For plane multigraphs, χmf is a.g., where m denotes either v or e. Quite possibly Conjectures 5.2, 5.3 are easily handled (or worse, silly). Moreover, as the author is unaware of the computational complexity of χvef , χvf , χef , it is not clear (to him) that they even share the same motivational basis as do Theorems 1.1, 3.1, 3.2. They are offered more for their potential to provide insight into Problem 5.4 than as a reflection of the author’s belief in their truth; indeed, they suggested themselves as this paper was being completed, and no serious attempts have been made to resolve any of our own conjectures. A glaring omission from our list of (multi)graph colouring invariants exhibiting a.g. behaviour is χ, whose determination is of course N P-hard (see [31] or, e.g., [13]). Thus, it is natural to pose the following (5.4) Problem Determine conditions on G to ensure that χ(G) is a.g.

(12)

Notice that two partial solutions to Problem 5.4 have already appeared in this paper. Theorem 3.1 shows that “G is the line graph of a multigraph” suffices for (12), while Theorem 1.1 offers the sufficient condition “G is the total graph of a (simple) graph”. It is intriguing to seek alternate sufficient conditions, or perhaps even a complete characterization of (12). We leave this for the reader to ponder. Acknowledgements The author thanks Jeannette Janssen, Jeff Kahn and Mike Molloy for making available as yet unpublished manuscripts. Thanks also to Karen Seyffarth, Marek Kubale and Glenn Hurlbert for assistance with some of the other references.

References [1] N. Alon, Restricted colorings of graphs, in “Surveys in Combinatorics, Keele, 1993” (K. Walker, Ed.), London Math. Soc. Lecture Note Ser. 187 (1993), 1–33. [2] L.D. Anderson, On edge-colourings of graphs, Math. Scand. 40 (1977), 161–175. [3] M. Behzad, Graphs and Their Chromatic Numbers, Dissertation, Michigan State University, East Lansing, MI, 1965. [4] C. Berge, Hypergraphs: Combinatorics of Finite Sets, NorthHolland, Amsterdam, 1989. 10

´s, Graph Theory: An Introductory Course, Springer[5] B. Bolloba Verlag, New York, 1979. ´s and A.J. Harris, List colorings of graphs, Graphs [6] B. Bolloba Combin. 1 (1985), 115–127. [7] O.V. Borodin, Structural theorem on plane graphs with application to the entire coloring number, J. Graph Theory 23 (1996), 233–239. ¨ggkvist, An improvement of Hind’s up[8] A. Chetwynd and R. Ha per bound on the total chromatic number, Combin. Probab. Comput. 5 (1996), 99–104. ´tal, Linear Programming, W.H. Freeman and Company, [9] V. Chva New York, 1983. [10] M.N. Ellingham and L. Goddyn, List edge colourings of some 1-factorable multigraphs, Combinatorica 16 (1996), 343–352. ˝ s, A. Rubin and H. Taylor, Choosability in graphs, [11] P. Erdo Congr. Numer. 26 (1979), 125–157. [12] F. Galvin, The list chromatic index of a bipartite multigraph, J. Combin. Theory Ser. B 63 (1995), 153–158. [13] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, San Francisco, 1979. [14] M.K. Goldberg, Multigraphs with a chromatic index that is nearly maximal (in Russian), Diskret. Analiz 23 (1973), 3–7. ¨ggkvist and J.C.M. Janssen, On the list-chromatic index [15] R. Ha of bipartite graphs, unpublished manuscript. ¨ggkvist and J.C.M. Janssen, New bounds on the list[16] R. Ha chromatic index of the complete graph, Combin. Probab. Comput., to appear. [17] H. Hind, An upper bound for the total chromatic number, Graphs Combin. 6 (1990), 153–159. [18] H. Hind, Recent developments in total colouring, Discrete Math. 125 (1994), 211–218. [19] H. Hind, M. Molloy and B. Reed, Total colouring with ∆ + poly(log ∆) colours, submitted.

11

[20] H. Hind, M. Molloy and B. Reed, Colouring a graph frugally, submitted. [21] I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981), 718–720. [22] T. Jensen and B. Toft, Graph Coloring Problems, Wiley, New York, 1995. [23] J. Kahn, Recent results on some not-so-recent hypergraph matching and covering problems, in “Extremal Problems for Finite Sets, Visegr´ ad, 1991” (P. Frankl, Z. F¨ uredi, G. Katona and D. Mikl´ os, Eds.), Bolyai Soc. Math. Studies 3 (1994), 305–353. [24] J. Kahn, Asymptotically good list-colorings, J. Combin. Theory Ser. A 73 (1996), 1–59. [25] J. Kahn, A linear programming perspective on the Frankl-R¨ odlPippenger theorem, Random Struct. Alg. 8 (1996), 149–157. [26] J. Kahn, Asymptotics of the chromatic index for multigraphs, J. Combin. Theory Ser. B 68 (1996), 233–254. [27] J. Kahn, Asymptotics of the list-chromatic index for multigraphs, submitted. [28] J. Kahn and P.M. Kayll, Fractional v. integral covers in hypergraphs of bounded edge size, J. Combin. Theory Ser. A, to appear. [29] J. Kahn and P.M. Kayll, On the stochastic independence properties of hard-core distributions, Combinatorica, to appear. [30] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373–395. [31] R.M. Karp, Reducibility among combinatorial problems, in “Complexity of Computer Computations” (R.E. Miller and J.W. Thatcher, Eds.), pp. 85–103, Plenum Press, New York, 1972. [32] P.M. Kayll, Asymptotically Good Covers in Hypergraphs, Dissertation, Rutgers University, New Brunswick, NJ, 1994. [33] P.M. Kayll, “Normal” distributions on matchings in a multigraph: overview with applications, Congr. Numer. 107 (1995), 179–191. [34] P.M. Kayll, Asymptotics of the total chromatic number for simple graphs, Technical Report #96-20, DIMACS, Piscataway, NJ, 1996.

12

[35] P.M. Kayll, Asymptotically good covers in hypergraphs, Diss. Summ. Math. 1 (1996), 9–16. [36] L.G. Khachian, A polynomial algorithm in linear programming (in Russian), Dokl. Acad. Nauk SSSR 244 (1979), 1093–1096 [English translation: Soviet Math. Dokl. 20 (1979), 191–194]. [37] K. Kilakos and B. Reed, Fractionally colouring total graphs, Combinatorica 13 (1993), 435–440. [38] M. Kubale, Some results concerning the complexity of restricted colorings of graphs, Discrete Appl. Math. 36 (1992), 35–46. ´sz and M.D. Plummer, Matching Theory, North-Holland, [39] L. Lova New York, 1986. [40] M. Molloy and B. Reed, A bound on the total chromatic number, Technical Report 304/96, Dept. of Computer Science, University of Toronto, 1996. [41] N. Pippenger and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combin. Theory Ser. A 51 (1989), 24–42. [42] J. Ryan, Fractional total colouring, Discrete Appl. Math. 27 (1990), 287–292. ´nchez-Arroyo, Determining the total colouring number is [43] A. Sa NP-hard, Discrete Math. 78 (1989), 315–319. [44] A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1986. [45] P.D. Seymour, Some unsolved problems on one-factorizations of graphs, in “Graph Theory and Related Topics, Waterloo, 1977” (J.A. Bondy and U.S.R. Murty, Eds.), pp. 367–368, Academic Press, New York, 1979. [46] T. Slivnik, Short proof of Galvin’s theorem on the list-chromatic index of a bipartite multigraph, Combin. Probab. Comput. 5 (1996), 91–94. [47] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz 3 (1964), 25–30. [48] V.G. Vizing, Some unsolved problems in graph theory, Russian Math Surveys 23 (1968), 125–141.

13

[49] V.G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Analiz 29 (1976), 3–10. [50] W. Wang, On the colorings of outerplanar graphs, Discrete Math. 147 (1995), 257–269. [51] H.P. Yap, Total Colourings of Graphs, Lecture Notes in Mathematics 1623, Springer, New York, 1996.

14