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A Local Epsilon Version of Reed's Conjecture
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Electronic Notes in Discrete Mathematics 61 (2017) 719–725 www.elsevier.com/locate/endm
A Local Epsilon Version of Reed’s Conjecture Tom Kelly 1 and Luke Postle 2,3 Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario, Canada
Abstract In 1998, Reed conjectured that every graph of maximum degree Δ and clique number ω can be colored with 12 (Δ + 1 + ω) colors, significantly strengthening Brooks’ Theorem. As evidence for his conjecture, he proved that this is true instead when the number of colors is some nontrivial convex combination of Δ + 1 and ω. In 1979, Erd˝os, Rubin, and Taylor proved that a connected graph G is L-colorable for every list-assignment L satisfying |L(v)| ≥ d(v) for all v ∈ V (G), unless every block of G is a clique or odd cycle. We ask if every graph G is L-colorable for every list-assignment L satisfying |L(v)| ≥ 12 (d(v) + 1 + ω(v)), where ω(v) denotes the size of the largest clique in G containing v. We prove that this is true instead when |L(v)| is some nontrivial convex combination of d(v) + 1 and ω(v), under certain mild assumptions. Keywords: Graph Theory, Coloring, List-Coloring, Reed’s Conjecture, Probabilistic Method
1 2 3
Email: [email protected] Email: [email protected] Partially supported by NSERC under Discovery Grant No. 2014-06162.
http://dx.doi.org/10.1016/j.endm.2017.07.028 1571-0653/© 2017 Published by Elsevier B.V.
720
1
T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
Introduction
Let G be a graph, and let L = (L(v) : v ∈ V (G)) be a collection of lists which we call available colors. If each set L(v) is non-empty, then we say that L is a list-assignment for G. If k is an integer and |L(v)| ≥ k for every v ∈ V (G), then we say that L is a k-list-assignment for G. An L-coloring of G is a mapping φ with domain V (G) such that φ(v) ∈ L(v) for every v ∈ V (G) and φ(u) = φ(v) for every pair of adjacent vertices u, v ∈ V (G). We say that a graph G is k-list-colorable, or k-choosable, if G has an L-coloring for every k-list-assignment L. If L(v) = {1, . . . , k} for every v ∈ V (G), then we call an L-coloring of G a k-coloring, and we say G is k-colorable if G has a k-coloring. The chromatic number of G, denoted χ(G), is the smallest k such that G is k-colorable. The list-chromatic number of G, denoted χ (G), is the smallest k such that G is k-list-colorable. A classical theorem of Brooks [2] states that the chromatic number of every connected graph G that is not a clique or an odd cycle is at most Δ(G), where Δ(G) denotes the maximum degree of G. This improves upon the trivial upper bound on the chromatic number of Δ(G) + 1. In 1998, Reed [10] conjectured that, up to rounding, the chromatic number of every graph G is at most the average of this trivial upper bound and the trivial lower bound of ω(G), where ω(G) denotes the size of the largest clique in G. Conjecture 1.1 (Reed’s Conjecture) [10] For every graph G, 1 χ(G) ≤ (Δ(G) + 1 + ω(G)) . 2 As evidence for his conjecture, Reed [10] proved it to be true for graphs of sufficiently large maximum degree and clique number. Theorem 1.2 [10] There exist constants Δ0 and ζ > 0 such that if G is a graph ofmaximum degree Δ(G) ≥ Δ0 and ω(G) ≥ (1 − ζ)(Δ(G) + 1), then χ(G) ≤ 12 (Δ(G) + 1 + ω(G)) . By combining Theorem 1.2 and Brooks’ Theorem with ε = min{Δ−1 0 , ζ/2}, Reed derived the following corollary. Corollary 1.3 [10] There exists ε > 0 such that every graph G satisfies χ(G) ≤ (1 − ε)(Δ(G) + 1) + εω(G). In this paper we are interested in list-coloring. It is natural to wonder if Brooks’ Theorem or even Reed’s Conjecture is true for the list-chromatic
T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
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number. In fact, in 1979, Erd˝os, Rubin, and Taylor [6] proved the following strengthening of Brooks’ Theorem. Theorem 1.4 [6] If G is a connected graph with list-assignment L such that for each v ∈ V (G), |L(v)| ≥ d(v), then G is L-colorable unless every block of G is a clique or odd cycle. We find that although Theorem 1.4 is well-known, there are too few theorems of a similar nature. We consider Theorem 1.4 to be the archetype of what we call a “local version”. In this paradigm, there are many new interesting questions that can be asked, such as the following “local version” of Reed’s conjecture for list-coloring. Conjecture 1.5 If G is a graph with list-assignment L such that for each v ∈ V (G), 1 |L(v)| ≥ (d(v) + 1 + ω(v)) , 2 where ω(v) is the size of the largest clique containing v, then G is L-colorable. Note that Conjecture 1.5, if true, would imply that Reed’s Conjecture is true even for list-coloring. We ask if Conjecture 1.5 holds when |L(v)| is instead some nontrivial convex combination of ω(v) and d(v) + 1 for every vertex v, i.e. is there a “local version” of Corollary 1.3. Our main result is the following, which says that this is true under some mild assumptions. 1 Theorem 1.6 Let ε ≤ 52 . If G is a graph of sufficiently large maximum degree and L is a list-assignment for G such that for all v ∈ V (G), |L(v)| ≥ ω(v) + log14 (Δ(G)) and
|L(v)| ≥ (1 − ε)(d(v) + 1) + εω(v), then G has an L-coloring. Note that Theorem 1.6 applies to all graphs of sufficiently large maximum degree Δ such that each vertex v satisfies d(v) ≥ ω(v) + (1 − ε)−1 log14 (Δ). Our next goal is to prove a version of Theorem 1.6 with no restriction on the relation between ω(v) and d(v). Improving the value of ε in Corollary 1.3 has attracted a lot of attention recently. For graphs of sufficiently large maximum degree, Reed proved that a value of ε = 1.4 · 10−8 suffices in Corollary 1.3, and he claimed that with a more careful analysis one could prove Corollary 1.3 with a value of ε = 2·10−4 . In 2016, King and Reed [9] provided a new proof of Corollary 1.3 that is much shorter. They did not try to optimize the value of ε, but their proof implies
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ε = 1.3 · 10−5 is true. Combining a result of Bruhn and Joos [3] from 2015 with the new proof of King and Reed yields ε = 830−1 . In 2017, Bonamy, Postle, and Perrett [1] announced a proof for ε = 26−1 , and Delcourt and Postle [5] announced a proof for ε = 13−1 . The result of Delcourt and Postle is actually proved for the list-chromatic number. With our current methods, we do not think we can improve upon the value of ε = 52−1 in Theorem 1.6 much further. Another avenue of research towards Reed’s Conjecture has centered around the following conjecture of King [7], which if true, implies Reed’s Conjecture. Conjecture 1.7 [7] For every graph G, 1 χ(G) ≤ max (d(v) + 1 + ω(v)) . v∈V (G) 2 King’s idea behind Conjecture 1.7 is that for certain classes of graphs it can be easier to prove than Reed’s Conjecture by using induction. Using this and the structure theory of claw-free graphs of Chudnovsky and Seymour, King [7] proved in 2009 that Reed’s Conjecture is true for claw-free graphs. The proof also appears in [8]. In 2013, Chudnovsky et. al. [4] proved that King’s Conjecture holds for quasi-line graphs, and in 2015 King and Reed [8] proved it for claw-free graphs with a three-colorable complement. Note that if true, Conjecture 1.5 implies Conjecture 1.7. It is natural to ask if Conjecture 1.7 holds when the maximum is taken instead over some nontrivial convex combination of d(v) + 1 and ω(v). This was not previously known, but under certain assumptions it is an application of our result, and it is even true for list-coloring. 1 Corollary 1.8 Let ε ≤ 52 . If G is a graph of sufficiently large maximum degree such that ω(G) ≤ (1 − ε)Δ(G) − log14 (Δ(G)), then
χ (G) ≤ max (1 − ε)(d(v) + 1) + εω(v). v∈V (G)
2
Proof Overview
There are two main steps to the proof of Theorem 1.6. The first is to prove that in a minimum counterexample every vertex satisfies some desirable structural property. The second step is to perform a random partial coloring of the graph and prove that with nonzero probability the partial coloring can be completed greedily.
T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
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The second step uses a variant of the naive coloring procedure, a widely studied technique. For each v ∈ V (G), we assign a color φ(v) ∈ L(v) uniformly at random. We say v is colored if it has no neighbors u such that φ(u) = φ(v) and |L(u)| ≤ |L(v)|. We introduced the requirement that |L(u)| ≤ |L(v)| for technical reasons concerning applications of Talagrand’s Inequality. Previous applications of this technique omit this requirement or introduce a coin flip to decide which of u and v to leave colored. We let G denote the graph induced by G on vertices that are not colored, and for each v ∈ V (G ) we let L (v) be the set of colors in L(v) that were not assigned to any colored neighbor of v. By the construction of G and L , if G has an L -coloring with non-zero probability, then G is L-colorable. We define three types of vertices. Definition 2.1 We say u ∈ N (v) is a subservient neighbor of v if |L(u)| < |L(v)|. We say v ∈ V (G) is lordly if the number of subservient neighbors is Ω(d(v) − ω(v)). Definition 2.2 We say u ∈ N (v) is an egalitarian neighbor of v if |L(v)| ≤ |L(u)| ≤ 1.4|L(v)|. We say v ∈ V (G) is egalitarian-sparse if the number of nonedges among its egalitarian neighbors is Ω(d(v)(d(v) − ω(v)). Definition 2.3 We say v ∈ V (G) is aberrant if |L(u)\L(v)| ≥ Ω(d(v) − ω(v)). |L(u)|
u∈N (v)
Note that if a vertex does not have many subservient or egalitarian neighbors, then it is aberrant. A vertex is also aberrant if it has Ω(d(v)) egalitarian neighbors u such that |L(u)| ≥ |L(v)| + Ω(d(v) − ω(v)). The notion of aberrance is useful because after an application of our naive random coloring procedure, an aberrant vertex v will have many colored neighbors u such that φ(u) ∈ / L(v). We can then prove using a modified version of Talagrand’s Inequality that if v is aberrant, P [|L (v)| > dG (v)] ≥ 1 −
Δ(G)−4 . 4
(1)
If a vertex v is egalitarian-sparse and not aberrant, then its egalitarian neighbors have many available colors in common. After an application of our naive random coloring procedure, an egalitarian-sparse vertex that is not aberrant will have many colors assigned to multiple colored neighbors. We can again prove using a modified version of Talagrand’s Inequality that (1)
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also holds if v is egalitarian-sparse. This idea has been used in [3,9,10] where such colors are called repeated colors. If a vertex v is lordly, we are unable to prove that (1) holds. To deal with this, we color the vertices of G in decreasing order by |L(v)|. Hence the subservient neighbors of a vertex come later in the ordering. We prove that a lordly vertex has many uncolored neighbors later in the ordering. In particular, if v ∈ V (G ) is lordly, P [|L (v)| > |{u ∈ N (v) ∩ V (G ) : |L(u)| ≥ |L(v)|}|] ≥ 1 −
Δ(G)−4 . 4
(2)
In order for (2) to hold, we use the common technique of equalizing coin flips, in which we uncolor some colored vertices with a certain probability so that each vertex is colored with some constant probability. From (1), (2), and the Lov´asz Local Lemma, we deduce the following. 1 . There exists Δ0 such that if Δ ≥ Δ0 and G is a Lemma 2.4 Let ε ≤ 52 graph with maximum degree Δ(G) ≤ Δ and a list-assignment L satisfying
|L(v)| ≥ (1 − ε)(d(v) + 1) + εω(v) for all v ∈ V (G), and if (i) |L(v)| ≥ ω(v) + log14 (Δ) and (ii) v is either lordly, egalitarian-sparse, or aberrant, for all v ∈ V (G), then G is L-colorable. Now the first step in the proof of Theorem 1.6 is to prove that in minimum counterexamples every vertex is either lordly, egalitarian-sparse, or aberrant. We say a graph G with list-assignment L is L-critical if G is not L-colorable but every proper induced subgraph of G is. Lemma 2.5 Let ε ≤ satisfying
1 . 52
If G is L-critical with respect to a list-assignment L
|L(v)| ≥ (1 − ε)(d(v) + 1) + εω(v) for all v ∈ V (G), then every vertex of G is either lordly, egalitarian-sparse, or aberrant. The proof of Theorem 1.6 now follows easily from Lemmas 2.4 and 2.5.
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References [1] Bonamy, M., T. Perrett and L. Postle, Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications, submitted. [2] Brooks, R. L., On colouring the nodes of a network, Mathematical Proceedings of the Cambridge Philosophical Society 37 (1941), p. 194–197. [3] Bruhn, H. and F. Joos, A stronger bound for the strong chromatic index, ArXiv e-prints (2015). [4] Chudnovsky, M., A. King, M. Plumettaz and P. Seymour, A local strengthening of Reed’s ω, Δ, χ conjecture for quasi-line graphs, SIAM J. Discrete Math. 27 (2013), pp. 95–108. [5] Delcourt, M. and L. Postle, On the List Coloring Version of Reed’s Conjecture, manuscript. [6] Erd˝os, P., A. L. Rubin and H. Taylor, Choosability in graphs, in: Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979), Congress. Numer., XXVI (1980), pp. 125–157. [7] King, A., “Claw-free graphs and two conjectures on omega, Delta, and chi,” ProQuest LLC, Ann Arbor, MI, 2009, 208 pp., thesis (Ph. D.)-McGill University (Canada). [8] King, A. and B. Reed, Claw-free graphs, skeletal graphs, and a stronger conjecture on ω, Δ, and χ, J. Graph Theory 78 (2015), pp. 157–194. [9] King, A. and B. Reed, A short proof that χ can be bounded ε away from Δ + 1 toward ω, J. Graph Theory 81 (2016), pp. 30–34. [10] Reed, B., ω, Δ, and χ, J. Graph Theory 27 (1998), pp. 177–212.
Electronic Notes in Discrete Mathematics 61 (2017) 719–725 www.elsevier.com/locate/endm
A Local Epsilon Version of Reed’s Conjecture Tom Kelly 1 and Luke Postle 2,3 Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario, Canada
Abstract In 1998, Reed conjectured that every graph of maximum degree Δ and clique number ω can be colored with 12 (Δ + 1 + ω) colors, significantly strengthening Brooks’ Theorem. As evidence for his conjecture, he proved that this is true instead when the number of colors is some nontrivial convex combination of Δ + 1 and ω. In 1979, Erd˝os, Rubin, and Taylor proved that a connected graph G is L-colorable for every list-assignment L satisfying |L(v)| ≥ d(v) for all v ∈ V (G), unless every block of G is a clique or odd cycle. We ask if every graph G is L-colorable for every list-assignment L satisfying |L(v)| ≥ 12 (d(v) + 1 + ω(v)), where ω(v) denotes the size of the largest clique in G containing v. We prove that this is true instead when |L(v)| is some nontrivial convex combination of d(v) + 1 and ω(v), under certain mild assumptions. Keywords: Graph Theory, Coloring, List-Coloring, Reed’s Conjecture, Probabilistic Method
1 2 3
Email: [email protected] Email: [email protected] Partially supported by NSERC under Discovery Grant No. 2014-06162.
http://dx.doi.org/10.1016/j.endm.2017.07.028 1571-0653/© 2017 Published by Elsevier B.V.
720
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T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
Introduction
Let G be a graph, and let L = (L(v) : v ∈ V (G)) be a collection of lists which we call available colors. If each set L(v) is non-empty, then we say that L is a list-assignment for G. If k is an integer and |L(v)| ≥ k for every v ∈ V (G), then we say that L is a k-list-assignment for G. An L-coloring of G is a mapping φ with domain V (G) such that φ(v) ∈ L(v) for every v ∈ V (G) and φ(u) = φ(v) for every pair of adjacent vertices u, v ∈ V (G). We say that a graph G is k-list-colorable, or k-choosable, if G has an L-coloring for every k-list-assignment L. If L(v) = {1, . . . , k} for every v ∈ V (G), then we call an L-coloring of G a k-coloring, and we say G is k-colorable if G has a k-coloring. The chromatic number of G, denoted χ(G), is the smallest k such that G is k-colorable. The list-chromatic number of G, denoted χ (G), is the smallest k such that G is k-list-colorable. A classical theorem of Brooks [2] states that the chromatic number of every connected graph G that is not a clique or an odd cycle is at most Δ(G), where Δ(G) denotes the maximum degree of G. This improves upon the trivial upper bound on the chromatic number of Δ(G) + 1. In 1998, Reed [10] conjectured that, up to rounding, the chromatic number of every graph G is at most the average of this trivial upper bound and the trivial lower bound of ω(G), where ω(G) denotes the size of the largest clique in G. Conjecture 1.1 (Reed’s Conjecture) [10] For every graph G, 1 χ(G) ≤ (Δ(G) + 1 + ω(G)) . 2 As evidence for his conjecture, Reed [10] proved it to be true for graphs of sufficiently large maximum degree and clique number. Theorem 1.2 [10] There exist constants Δ0 and ζ > 0 such that if G is a graph ofmaximum degree Δ(G) ≥ Δ0 and ω(G) ≥ (1 − ζ)(Δ(G) + 1), then χ(G) ≤ 12 (Δ(G) + 1 + ω(G)) . By combining Theorem 1.2 and Brooks’ Theorem with ε = min{Δ−1 0 , ζ/2}, Reed derived the following corollary. Corollary 1.3 [10] There exists ε > 0 such that every graph G satisfies χ(G) ≤ (1 − ε)(Δ(G) + 1) + εω(G). In this paper we are interested in list-coloring. It is natural to wonder if Brooks’ Theorem or even Reed’s Conjecture is true for the list-chromatic
T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
721
number. In fact, in 1979, Erd˝os, Rubin, and Taylor [6] proved the following strengthening of Brooks’ Theorem. Theorem 1.4 [6] If G is a connected graph with list-assignment L such that for each v ∈ V (G), |L(v)| ≥ d(v), then G is L-colorable unless every block of G is a clique or odd cycle. We find that although Theorem 1.4 is well-known, there are too few theorems of a similar nature. We consider Theorem 1.4 to be the archetype of what we call a “local version”. In this paradigm, there are many new interesting questions that can be asked, such as the following “local version” of Reed’s conjecture for list-coloring. Conjecture 1.5 If G is a graph with list-assignment L such that for each v ∈ V (G), 1 |L(v)| ≥ (d(v) + 1 + ω(v)) , 2 where ω(v) is the size of the largest clique containing v, then G is L-colorable. Note that Conjecture 1.5, if true, would imply that Reed’s Conjecture is true even for list-coloring. We ask if Conjecture 1.5 holds when |L(v)| is instead some nontrivial convex combination of ω(v) and d(v) + 1 for every vertex v, i.e. is there a “local version” of Corollary 1.3. Our main result is the following, which says that this is true under some mild assumptions. 1 Theorem 1.6 Let ε ≤ 52 . If G is a graph of sufficiently large maximum degree and L is a list-assignment for G such that for all v ∈ V (G), |L(v)| ≥ ω(v) + log14 (Δ(G)) and
|L(v)| ≥ (1 − ε)(d(v) + 1) + εω(v), then G has an L-coloring. Note that Theorem 1.6 applies to all graphs of sufficiently large maximum degree Δ such that each vertex v satisfies d(v) ≥ ω(v) + (1 − ε)−1 log14 (Δ). Our next goal is to prove a version of Theorem 1.6 with no restriction on the relation between ω(v) and d(v). Improving the value of ε in Corollary 1.3 has attracted a lot of attention recently. For graphs of sufficiently large maximum degree, Reed proved that a value of ε = 1.4 · 10−8 suffices in Corollary 1.3, and he claimed that with a more careful analysis one could prove Corollary 1.3 with a value of ε = 2·10−4 . In 2016, King and Reed [9] provided a new proof of Corollary 1.3 that is much shorter. They did not try to optimize the value of ε, but their proof implies
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T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
ε = 1.3 · 10−5 is true. Combining a result of Bruhn and Joos [3] from 2015 with the new proof of King and Reed yields ε = 830−1 . In 2017, Bonamy, Postle, and Perrett [1] announced a proof for ε = 26−1 , and Delcourt and Postle [5] announced a proof for ε = 13−1 . The result of Delcourt and Postle is actually proved for the list-chromatic number. With our current methods, we do not think we can improve upon the value of ε = 52−1 in Theorem 1.6 much further. Another avenue of research towards Reed’s Conjecture has centered around the following conjecture of King [7], which if true, implies Reed’s Conjecture. Conjecture 1.7 [7] For every graph G, 1 χ(G) ≤ max (d(v) + 1 + ω(v)) . v∈V (G) 2 King’s idea behind Conjecture 1.7 is that for certain classes of graphs it can be easier to prove than Reed’s Conjecture by using induction. Using this and the structure theory of claw-free graphs of Chudnovsky and Seymour, King [7] proved in 2009 that Reed’s Conjecture is true for claw-free graphs. The proof also appears in [8]. In 2013, Chudnovsky et. al. [4] proved that King’s Conjecture holds for quasi-line graphs, and in 2015 King and Reed [8] proved it for claw-free graphs with a three-colorable complement. Note that if true, Conjecture 1.5 implies Conjecture 1.7. It is natural to ask if Conjecture 1.7 holds when the maximum is taken instead over some nontrivial convex combination of d(v) + 1 and ω(v). This was not previously known, but under certain assumptions it is an application of our result, and it is even true for list-coloring. 1 Corollary 1.8 Let ε ≤ 52 . If G is a graph of sufficiently large maximum degree such that ω(G) ≤ (1 − ε)Δ(G) − log14 (Δ(G)), then
χ (G) ≤ max (1 − ε)(d(v) + 1) + εω(v). v∈V (G)
2
Proof Overview
There are two main steps to the proof of Theorem 1.6. The first is to prove that in a minimum counterexample every vertex satisfies some desirable structural property. The second step is to perform a random partial coloring of the graph and prove that with nonzero probability the partial coloring can be completed greedily.
T. Kelly, L. Postle / Electronic Notes in Discrete Mathematics 61 (2017) 719–725
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The second step uses a variant of the naive coloring procedure, a widely studied technique. For each v ∈ V (G), we assign a color φ(v) ∈ L(v) uniformly at random. We say v is colored if it has no neighbors u such that φ(u) = φ(v) and |L(u)| ≤ |L(v)|. We introduced the requirement that |L(u)| ≤ |L(v)| for technical reasons concerning applications of Talagrand’s Inequality. Previous applications of this technique omit this requirement or introduce a coin flip to decide which of u and v to leave colored. We let G denote the graph induced by G on vertices that are not colored, and for each v ∈ V (G ) we let L (v) be the set of colors in L(v) that were not assigned to any colored neighbor of v. By the construction of G and L , if G has an L -coloring with non-zero probability, then G is L-colorable. We define three types of vertices. Definition 2.1 We say u ∈ N (v) is a subservient neighbor of v if |L(u)| < |L(v)|. We say v ∈ V (G) is lordly if the number of subservient neighbors is Ω(d(v) − ω(v)). Definition 2.2 We say u ∈ N (v) is an egalitarian neighbor of v if |L(v)| ≤ |L(u)| ≤ 1.4|L(v)|. We say v ∈ V (G) is egalitarian-sparse if the number of nonedges among its egalitarian neighbors is Ω(d(v)(d(v) − ω(v)). Definition 2.3 We say v ∈ V (G) is aberrant if |L(u)\L(v)| ≥ Ω(d(v) − ω(v)). |L(u)|
u∈N (v)
Note that if a vertex does not have many subservient or egalitarian neighbors, then it is aberrant. A vertex is also aberrant if it has Ω(d(v)) egalitarian neighbors u such that |L(u)| ≥ |L(v)| + Ω(d(v) − ω(v)). The notion of aberrance is useful because after an application of our naive random coloring procedure, an aberrant vertex v will have many colored neighbors u such that φ(u) ∈ / L(v). We can then prove using a modified version of Talagrand’s Inequality that if v is aberrant, P [|L (v)| > dG (v)] ≥ 1 −
Δ(G)−4 . 4
(1)
If a vertex v is egalitarian-sparse and not aberrant, then its egalitarian neighbors have many available colors in common. After an application of our naive random coloring procedure, an egalitarian-sparse vertex that is not aberrant will have many colors assigned to multiple colored neighbors. We can again prove using a modified version of Talagrand’s Inequality that (1)
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also holds if v is egalitarian-sparse. This idea has been used in [3,9,10] where such colors are called repeated colors. If a vertex v is lordly, we are unable to prove that (1) holds. To deal with this, we color the vertices of G in decreasing order by |L(v)|. Hence the subservient neighbors of a vertex come later in the ordering. We prove that a lordly vertex has many uncolored neighbors later in the ordering. In particular, if v ∈ V (G ) is lordly, P [|L (v)| > |{u ∈ N (v) ∩ V (G ) : |L(u)| ≥ |L(v)|}|] ≥ 1 −
Δ(G)−4 . 4
(2)
In order for (2) to hold, we use the common technique of equalizing coin flips, in which we uncolor some colored vertices with a certain probability so that each vertex is colored with some constant probability. From (1), (2), and the Lov´asz Local Lemma, we deduce the following. 1 . There exists Δ0 such that if Δ ≥ Δ0 and G is a Lemma 2.4 Let ε ≤ 52 graph with maximum degree Δ(G) ≤ Δ and a list-assignment L satisfying
|L(v)| ≥ (1 − ε)(d(v) + 1) + εω(v) for all v ∈ V (G), and if (i) |L(v)| ≥ ω(v) + log14 (Δ) and (ii) v is either lordly, egalitarian-sparse, or aberrant, for all v ∈ V (G), then G is L-colorable. Now the first step in the proof of Theorem 1.6 is to prove that in minimum counterexamples every vertex is either lordly, egalitarian-sparse, or aberrant. We say a graph G with list-assignment L is L-critical if G is not L-colorable but every proper induced subgraph of G is. Lemma 2.5 Let ε ≤ satisfying
1 . 52
If G is L-critical with respect to a list-assignment L
|L(v)| ≥ (1 − ε)(d(v) + 1) + εω(v) for all v ∈ V (G), then every vertex of G is either lordly, egalitarian-sparse, or aberrant. The proof of Theorem 1.6 now follows easily from Lemmas 2.4 and 2.5.
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