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Planar graphs with girth 20 are additively 3-choosable
Discrete Applied Mathematics xxx (xxxx) xxx
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Planar graphs with girth 20 are additively 3-choosable ∗
Axel Brandt a , , Nathan Tenpas b , Carl R. Yerger c a
Northern Kentucky University, Highland Heights, KY 41099, United States of America Vanderbilt University, Nashville, TN 37235, United States of America c Davidson College, Davidson, NC 28035, United States of America b
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Article history: Received 17 May 2018 Received in revised form 20 June 2019 Accepted 28 August 2019 Available online xxxx Keywords: Additive coloring Lucky labeling Reducible configurations Discharging method Combinatorial Nullstellensatz
a b s t r a c t The additive choice number of a graph G, denoted by chΣ (G), is the minimum positive integer k such that, whenever each vertex is given a list of at least k positive integers, vertex labels can be chosen from respective lists in such a way that adjacent vertices have distinct sums of labels on their neighbors. Recently, bounds on the additive choice number have been proven for planar graphs under certain girth assumptions. In this paper, we prove that chΣ (G) ≤ 3 for the planar graph G with girth at least 20. Our approach uses the Combinatorial Nullstellensatz to streamline arguments within a proof using the discharging method. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we consider only finite, undirected graphs. We abuse notation slightly by assuming a fixed plane embedding of a planar graph. When not otherwise specified, see [11] for notation. An additive coloring of a graph G is a labeling of the vertices of G with natural numbers such that two adjacent vertices have distinct sums of labels on their neighbors. The additive coloring number of G, denoted by χΣ (G), is defined to be the minimum natural number k for which G has an additive coloring using labels in {1, . . . , k}. We briefly mention that additive coloring was introduced in the literature as lucky labeling by Czerwiński, Grytczuk, and Żelazny [10]. It has also been referred to as open distinguishing by Axenovich et al. [4]. We direct the interested reader to [8] for a more detailed exposition of results in this area. In 2009, Czerwiński, Grytczuk, and Żelazny conjectured the following: Conjecture 1.1 ([10]). For every graph G, χΣ (G) ≤ χ (G). This conjecture is open in general, and χΣ (Kn ) = χ (Kn ) implies that the conjecture would be best possible if true. Of particular interest, there is currently no known constant bound for the class of bipartite graphs. The best known bound for the class of planar graphs is given by the following: Theorem 1.2 ([5]). If G is a planar graph, then χΣ (G) ≤ 468. Recently, significant progress has been made for planar graphs under certain girth assumptions. The girth of a graph G, denoted by girth(G), is the length of the shortest cycle in G. Bartnicki et al. [5] use the existence of an I,F-partition, which we do not discuss here, to show that χΣ (G) ≤ 4 for planar graphs G with girth at least 13. Using an improved result on the existence of I,F-partitions [6], their result was later improved as follows: ∗ Corresponding author. E-mail addresses: [email protected] (A. Brandt), [email protected] (N. Tenpas), [email protected] (C.R. Yerger). https://doi.org/10.1016/j.dam.2019.08.021 0166-218X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Fig. 1. Configurations that do not appear in G: P(1, 0, 1), P(1, 1, 1), P(1, 1, 0, 0), P(0, 1, 0, 0), P(1, 0, 0, 0), and P(0, 0, 0, 0, 0).
Theorem 1.3 ([8]). If G is a planar graph with girth at least 10, then χΣ (G) ≤ 4. A list version of additive coloring has been studied, and similar results have been obtained. A graph G is additively k-choosable if an additive coloring can be selected from any list assignment of natural numbers to the vertices of G where each list has at least k distinct elements. The additive choice number of G, denoted by chΣ (G), is the minimum natural number k such that G is additively k-choosable. Ahadi and Dehghan [1] have shown that χΣ (G) and chΣ (G) can be arbitrarily far apart. Theorem 1.4 ([2]). If G is a forest, then chΣ (G) ≤ 3. Theorem 1.5 ([8]). Let G be a planar graph with girth g. If g ≥ 5, then chΣ (G) ≤ 19. If g ≥ 6, then chΣ (G) ≤ 9. If g ≥ 7, then chΣ (G) ≤ 8. If g ≥ 26, then chΣ (G) ≤ 3. In [7] the authors provide a shorter proof for the girth 26 result of Theorem 1.5. Expanding on that approach and incorporating aspects of the original proof enables the result to be strengthened as follows: Theorem 1.6.
If G is a planar graph with girth at least 20, then chΣ (G) ≤ 3.
2. Notation and tools As standard notation for proofs utilizing the discharging method, a j-, j− -, or j+ -vertex is a vertex with degree j, at most j, or at least j, respectively. Similarly, a j-, j− -, or j+ -neighbor of v is a j-, j− -, or j+ -vertex, respectively, adjacent to v . For the remainder of this paper, suppose G is a vertex-minimum counterexample to Theorem 1.6. That is, let G be a planar graph with girth at least 20 and chΣ (G) ≥ 4 such that chΣ (H) ≤ 3 for any graph H with |V (H)| < |V (G)|. By the nature of additive coloring, G is connected. To show that G does not exist, we utilize the following from [8] as it applies to our main result, that is when k = 3 and Q = ∅: Lemma 2.1 ([8], Lemma 3.1 Part b). A vertex v in G with r 1-neighbors satisfies 1 + 2r ≤ d(v ). For v ∈ V (G) and S ⊆ V (G), let dist(v, S) be the minimum distance from v to some vertex in S. As such, for a vertex v , define the cycle distance of v , denoted by cdist(v ), to be the minimum distance from v to a cycle of G, that is cdist(v ) := min{dist(v, V (C )): C is a cycle in G}. Note that this definition is well-defined because Theorem 1.4 implies G contains a cycle. The following appears in [7], and we include the proof for completeness. Lemma 2.2 ([7]). Let v be a 1-vertex in G. Then cdist(v ) = 1. Proof. Assume cdist(v ) ≥ cdist(u) for every 1-vertex u in G. Toward a contradiction, suppose cdist(v ) ≥ 2. Let w be the neighbor of v . By the choice of v , d(w ) ≥ 2 and w has exactly one 2+ -neighbor. Hence, w has (d(w ) − 1) 1-neighbors. Since 1 + 2(d(w ) − 1) = 2d(w ) − 1 ≥ d(w ) + 1 > d(w ), Lemma 2.1 implies that w does not exist in G, a contradiction. □ The configurations in Fig. 1 do not appear in G by Lemma 3.8 in [8]. Toward describing these and similar configurations, let P(t2 , . . . , tn−1 ) be the path v1 · · · vn such that, for each i in {2, . . . , n − 1}, the vertex vi has ti 1-neighbors and d(vi ) = 2 + ti . Note that Lemma 2.1 implies that ti ≤ 1 for each such vi . The following appears in [8] without proof, which we provide here for completeness. Lemma 2.3 ([8]). Any P(t2 , . . . , tn−1 ) in G satisfies n ≤ 6. Proof. The following table lists all possibilities for P(t2 , . . . , t6 ) along with the configurations pictured in Fig. 1 that preclude their existence in G. Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Fig. 2. An image of P(1, 1, 0) ⊙ P(1, 0, 0, 1) with respective underlying paths v1 · · · v5 and v1′ · · · v6′ .
Fig. 3. A visual depiction of G where the ellipses represent the rest of G, each of which includes a cycle containing u, x, y, and z, respectively, and are only connected by paths containing at least three labeled vertices.
Configuration
Forbids P(t2 , . . . , t6 )
P(1, 0, 1)
00101, 10110, 00111, 00011, 00010, 00001, 00000
P(1, 1, 1) P(1, 1, 0, 0) P(0, 1, 0, 0) P(1, 0, 0, 0) P(0, 0, 0, 0, 0)
01010, 10111, 01110, 00110, 00100, 10000,
01011, 11010, 01111, 01100, 01000, 10001
01101, 11011, 11100, 10011, 01001,
10100, 10101, 11101 11110, 11111 11000, 11001 10010
This list suffices because any P(t2 , . . . , tn−1 ) with n > 6 would necessarily contain one of the P(t2 , . . . , t6 ) listed above.
□
For P(t2 , . . . , tm−1 ) with underlying path v1 · · · vm , let P(t2 , . . . , tn−1 ) ⊙ P(t2 , . . . , tm−1 ) denote the graph obtained from ′ ′ P(t2 , . . . , tn−1 ) and P(t2′ , . . . , tm −1 ) by identifying vn and v1 . See Fig. 2 for an example of this construction. As in [7], define the thread degree of a vertex v , denoted by d2+ (v ), as the number of 2+ -neighbors of v . A hub vertex is a vertex v with d2+ (v ) ≥ 3, and a stable vertex is a hub vertex contained in a cycle. Define a (u, v )-thread to be a path whose endpoints u and v are hub vertices, and whose internal vertices are 2-vertices. In Fig. 3, all labeled vertices are hubs, but v and w are not stable. Note that Lemma 2.2 implies that the set of 2-vertices of G is a subset of the set of internal vertices of (u, v )-threads in G. Also, every 1-vertex of G is either adjacent to a hub vertex, or in some P(t2 , . . . , tn−1 ). ′
′
′
′
′
′
3. Some additional structure In this section we describe additional structural properties of G. Specifically, we enumerate some configurations that do not appear in G. We will utilize the following result of Alon: Theorem 3.1 field F. If there is a ∏ (Combinatorial ∑ Nullstellensatz [3]). Let f be a polynomial of degree t in m variables over a ∏ t monomial xi i in f with ti = t whose coefficient is nonzero in F, then f is nonzero at some point of Ti , where each Ti is a set of ti + 1 distinct values in F. We follow the approach of [8], which builds a polynomial whose nonzero solutions satisfy the restrictions for additive coloring. We include the following as an example. Lemma 3.2.
The configurations P(1, 1, 0) and P(1, 0, 0, 1) do not appear in G.
Proof. Let L be a function that assigns every vertex of G a list of three natural numbers. Suppose that P(1, 1, 0) appears in G. Let v1 · · · v5 be the underlying path of P(1, 1, 0) and let p2 and p3 be the pendant vertices adjacent to v2 and v3 , respectively. Let G′ = G − {p2 , p3 , v3 }. By the minimality of G, chΣ (G′ ) ≤ 3. As such, let ℓ be an additive coloring from L on V (G′ ). We will use Theorem 3.1 to guarantee that there exist ℓ(p2 ) ∈ L(p2 ), ℓ(p3 ) ∈ L(p3 ), and ℓ(v3 ) ∈ L(v3 ) that extend ℓ to be an additive coloring of G. Notice that the only edges impacted by the labels of p2 , p3 , and v3 are vi vi+1 for i ∈ {1, 2, 3, 4}, v2 p2 , and v3 p3 . The following polynomial has variables x, y, and z corresponding to the choice of labels at p2 , p3 , and v3 , respectively, and Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Fig. 4. A picture of P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ , t4′ ), where dash-dot edges indicate the potential of that edge’s existence.
has factors corresponding to a difference of label sums at endpoints of the edges v1 v2 , v2 p2 , v2 v3 , v3 p3 , v3 v4 , and v4 v5 , respectively: f (x, y, z) = (x + z + ℓ(v1 ) − S(v1 )) × (ℓ(v1 ) + x + z − ℓ(v2 ))
× (ℓ(v1 ) + x + z − ℓ(v2 ) − y − ℓ(v4 )) × (ℓ(v2 ) + y + ℓ(v4 ) − z) × (ℓ(v2 ) + y + ℓ(v4 ) − z − ℓ(v5 )) × (z + ℓ(v5 ) − S(v5 )), where S(v ) = u∈N(v ) ℓ(u). Notice that each factor of f is linear with respect to some variable. Thus, selecting a constant from a factor when expanding f causes the degree of the monomial obtained to be less than the degree of f . As such, the coefficient of x2 y2 z 2 in f (x, y, z) is equal to its coefficient in (x + z)2 (x + z − y)(y − z)2 z, which is 5. By Theorem 3.1, there is a choice of ℓ(p2 ) ∈ L(p2 ), ℓ(p3 ) ∈ L(p3 ), and ℓ(v3 ) ∈ L(v3 ) that is a nonzero solution of f . These labels then extend ℓ to an additive coloring of G. Hence, chΣ (G) ≤ 3, a contradiction with the choice of G. The proof for P(1, 0, 0, 1) is similar. Let v1 · · · v6 be the underlying path of P(1, 0, 0, 1) and let p2 and p5 be the pendant vertices adjacent to v2 and v5 , respectively. Let G′ = G − {p2 , v3 , v4 , p5 }. The following polynomial has variables w, x, y, and z corresponding to the choice of labels at p2 , v3 , v4 , and p5 and factors corresponding to the edges v1 v2 , v2 p2 , v2 v3 , v3 v4 , v4 v5 , v5 p5 , and v5 v6 , respectively:
∑
g(w, x, y, z) = (ℓ(v1 ) + w + x − S(v1 )) × (ℓ(v1 ) + w + x − ℓ(v2 ))
× (ℓ(v1 ) + w + x − ℓ(v2 ) − y) × (ℓ(v2 ) + y − x − ℓ(v5 )) × (x + ℓ(v5 ) − y − z − ℓ(v6 )) × (y + z + ℓ(v6 ) − ℓ(v5 )) × (y + z + ℓ(v6 ) − S(v6 )), where S(v ) = u∈N(v ) ℓ(u). As before, we may omit constants without altering the coefficient of maximum degree terms. The coefficient of w 2 x2 y2 z in g(w, x, y, z) is equal to its coefficient in (w + x)2 (w + x − y)(y − x)(x − y − z)(y + z)2 , which is 17. □
∑
The following lemmas are obtained using Mathematica to loop through all possible configurations. See the Appendix for relevant code. For an intuition into coding polynomials that allow for the possible nonexistence of edges, we use the configuration in Fig. 4 as an example. For this example, we will abuse notation by using vi , pi , vi as both names of vertices and the names for corresponding variables in the polynomial. We delete the labeled vertices (where each pi may or may not initially exist). Let Pi and Si be {0, 1}-indicators for the existence of each pi and si , respectively. Our code loops through all 27+2 possible indicator assignments, which cover all potential subconfigurations. These indicator assignments are used as exponents on factors and coefficients on variables in sums. For example, the factor corresponding to the edge v3 p3 is (v2 + P3 p3 + v4 − v3 )P3 . Toward verifying the polynomials cover all possibilities, note that Lemma 2.1 implies that vertices with thread degree 3 have at most two 1-neighbors. Lemma 3.3. Let v1 · · · v13 be the underlying path of P(t2 , t3 , t4 ) ⊙ P(t2′ , t3′ , t4′ ) ⊙ P(t2′′ , t3′′ , t4′′ ). If d2+ (v5 ) = d2+ (v9 ) = 3, then P(t2 , t3 , t4 ) ⊙ P(t2′ , t3′ , t4′ ) ⊙ P(t2′′ , t3′′ , t4′′ ) does not appear in G. Lemma 3.4. Let v1 · · · v10 be the underlying path of P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ , t4′ ). If d2+ (v6 ) = 3, then P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ , t4′ ) does not appear in G. Lemma 3.5. Let v1 · · · v13 be the underlying path of P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ ) ⊙ P(t2′′ , t3′′ , t4′′ ). If d2+ (v6 ) = d2+ (v9 ) = 3, then P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ ) ⊙ P(t2′′ , t3′′ , t4′′ ) does not appear in G. 4. Proof of Theorem 1.6 We construct a graph G′ from G and use a discharging argument to show that G′ , and thus G, does not exist. Our construction of G′ from G follows these steps in order: Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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(S1) Delete all 1-vertices of G. (S2) For all (u, v )-threads with a cut-edge, contract all edges of the (u, v )-thread. (S3) Replace all remaining (u, v )-threads with the edge uv . Recall that G is simple, planar, connected, and has girth at least 20. Using Lemma 2.3, it is straightforward to show that G′ is simple, planar, and connected with δ (G′ ) ≥ 3 and girth(G′ ) ≥ 4. Thus G′ is also 2-edge-connected. Proposition 3.1 in [9] gives
∑
(d(v ) − 4) +
v∈V (G′ )
∑
(l(f ) − 4) = −8.
f ∈F (G′ )
Thus assigning each vertex v an initial charge µ(v ) = d(v ) − 4 and each face f an initial charge µ(f ) = l(f ) − 4 gives an initial total charge of −8. Consider distributing charge from faces to vertices as follows: (R1) Each 3-vertex receives
1 3
charge from each of its incident faces.
We will show that distributing charge in this way results in every vertex and face of G′ having a nonnegative final charge, for a total charge at least 0, which is an impossibility. This impossibility implies that G′ does not exist, and, by construction, that G does not exist. Hence, no counterexample exists to Theorem 1.6. Every Vertex of G′ has Nonnegative Final Charge We first consider the final charge of a vertex v . If d(v ) ≥ 4, then v does not lose or gain any charge under (R1). Thus v would have final charge µ(v ) = d(v ) − 4 ≥ 0. Since δ (G′ ) ≥ 3, it remains to consider when d(v ) = 3. In this case, the 2-edge-connectivity of G′ implies that v is incident to 3 faces. Thus v has final charge µ(v ) + 3( 31 ) = (3 − 4) + 1 = 0. Therefore, all vertices of G′ have nonnegative final charge. Every Face of G′ has Nonnegative Final Charge We now turn our attention to the faces of G′ . Let f be a face of G′ . If f is incident to t 3-vertices, then f has final charge l(f ) − 4 − t( 31 ) ≥ 23 l(f ) − 4. Thus if l(f ) ≥ 6, then f has nonnegative final charge. Since girth(G′ ) ≥ 4, it remains to consider when l(f ) ∈ {4, 5}. From the previous calculation, it suffices to show that f is incident to four 4+ -vertices when l(f ) = 4 and to show that f is incident to at least two 4+ -vertices when l(f ) = 5. In constructing G′ from G, (S3) replaces each (u, v )-thread with an edge. We say that (E1 , E2 , . . . , El(f ) ) is a replacement sequence for a face f if the edges incident to f , listed consecutively, were obtained during (S3) by replacing Ei vertices in a (u, v )-thread. Let l(f ) = 4. Since girth(G) ≥ 20, there must be at least 16 internal vertices of (u, v )-threads replaced during (S3). Lemma 2.3 implies that each edge in the boundary of f was obtained during (S3) from a (u, v )-thread with at most 4 internal vertices. Thus f is a (4, 4, 4, 4)-face. By Lemma 3.4, all four vertices incident to f are 4+ -vertices, as desired. Let l(f ) = 5. Since girth(G) ≥ 20, there must be at least 15 internal vertices of (u, v )-threads replaced during (S3). Lemma 2.3 implies that there are at most 20 internal vertices of (u, v )-threads replaced during (S3). We consider all of the ways to distribute these vertices to edges in the boundary of f , and will enumerate the cases based on the number 4s in the replacement sequence of f . Let the replacement sequence of f have no 4. Then f is a (3, 3, 3, 3, 3) face. Two applications of Lemma 3.3 imply that f is incident at least two 4+ -vertices, as desired. Let the replacement sequence of f have exactly one 4. Since there are either 11 or 12 vertices replaced by the other edges of f , the replacement sequence has at least three 3s and at most one 2. Thus the replacement sequence of f , up to symmetry, is (4, 3, 3, 3, 3), (4, 3, 3, 2, 3), or (4, 3, 3, 3, 2). For the first two possible replacement sequences of f , two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. For the last possible replacement sequence of f , applications of Lemmas 3.3 and 3.4 imply that f is incident to at least two 4+ -vertices, as desired. Let the replacement sequence of f have exactly two 4s. First, assume that the two 4s are consecutive. If (4, 4, 3) is a subsequence of the replacement sequence of f , then two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. Otherwise, the replacement sequence of f is (4, 4, 2, 3, 2) and applications of Lemmas 3.4 and 3.5 imply that f is incident to at least two 4+ -vertices. Now, assume that the two 4s are not consecutive. If either (4, 3, 4) or (4, 3, 3, 4) is a subsequence of the replacement sequence of f , then two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. Otherwise, (4, 2, 4, 3) is a subsequence of the replacement sequence of f and applications of Lemmas 3.4 and 3.5 imply that f is incident to at least two 4+ -vertices, as desired. Finally, let the replacement sequence of f have at least three 4s. If (4, 4, 4) is a subsequence of the replacement sequence of f , then two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. Otherwise, (4, 4, 3, 4) or (4, 4, 2, 4) is a subsequence of the replacement sequence of f , and applications of Lemma 3.4, and Lemma 3.4 or 3.5, respectively, imply that f is incident to at least two 4+ -vertices, as desired. Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Appendix
Mathematica code for Lemma 3.3
(* 3-3-3 is 3-reducible - cycle through all possible combinations *) coef = Table[0, {i, 2^13}]; Do[ (* 13 independent choices for pendants - 9 p, 4 s *) {P1, P2, P3, P4, P5, P6, P7, P8, P9, S1, S2, S3, S4} = IntegerDigits[n, 2, 13]; (* 0-1 indicator for pendants by name *) (* function obtained by deleting v2-v8 and any pendants that exist *) f[v2_, v3_, v4_, v5_, v6_, v7_, v8_, p1_, p2_, p3_, p4_, p5_, p6_, p7_, p8_, p9_, s1_, s2_, s3_, s4_] := ( (* internal edges left to right - P# p# includes pendant if exists *) (P1 p1 + v2) (P1 p2 + v2 - P2 p2 - v3) (P2 p2 + v3 - v2 P3 p3) (v2 + P3 p3 - v3 - S1 s1 - S2 s2 - v4) (v3 + S1 s1 + S2 s2 + v4 - P4 p4 - v5) (P4 p4 + v5 - v4 - P5 p5 - v6) (v4 + P5 p5 + v6 - v5 - P6 p6) (v5 + P6 p6 - v6 - S3 s3 - S4 s4 v7) (v6 + S3 s3 + S4 s4 + v7 - P7 p7 - v8) (P7 p7 + v8 - v7 P8 p8) (v7 + P8 p8 - v8 - P9 p9) (v8 + P9 p9) (* pendant edges left to right - ^P# includes factor if edge exists *) (P1 p1 + v2)^P1 (P2 p2 + v3 - v2)^P2 (v2 + P3 p3 - v3)^ P3 (v3 + S1 s1 + S2 s2 + v4)^(S1 + S2) (P4 p4 + v5 - v4)^ P4 (v4 + P5 p5 + v6 - v5)^P5 (v5 + P6 p6 - v6)^ P6 (v6 + S3 s3 + S4 s4 + v7)^(S3 + S4) (P7 p7 + v8 - v7)^ P7 (v7 + P8 p8 - v8)^P8 (v8 + P9 p9)^P9 (* 3rd thread edge from stable vertices *) (v3 + S1 s1 + S2 s2 + v4) (v6 + S3 s3 + S4 s4 + v7) ); (* track the desired term in polynomial for Nullstellensatz *) coef[[n + 1]] = Coefficient[ f[v2, v3, v4, v5, v6, v7, v8, p1, p2, p3, p4, p5, p6, p7, p8, p9, s1, s2, s3, s4], (* base vertices *) v2^2 v3^2 v4^2 v5^2 v6^2 v7^2 v8^2 (* pendants *) p1^P1 p2^P2 p3^P3 p4^P4 p5^P5 p6^P6 p7^P7 p8^P8 p9^P9 (* stable vtx *) s1^S1 s2^S2 s3^S3 s4^S4]; (* check for success *) If[coef[[n + 1]] == 0, Print["Failure with n=", n]; Break, If[n == 2^13 - 1, Print["Success!"] (* Display coefficients *) Print[coef], Continue ] ] , {n, 0, 2^13 - 1}] Mathematica code for Lemma 3.4
(* 4-3 is 3-reducible - cycle through all possible combinations *) coef = Table[0, {i, 2^9}]; Do[ (* 9 independent choices for pendants - 7 p, 2 s *) {P1, P2, P3, P4, P5, P6, P7, S1, S2} = IntegerDigits[n, 2, 9]; (* indicator for pendants by name *) (* function obtained by deleting v2-v8 and any pendants *) f[v2_, v3_, v4_, v5_, v6_, p1_, p2_, p3_, p4_, p5_, p6_, p7_, s1_, s2_] := ( (* internal edges left to right - P# p# includes pendant if exists *) Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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(P1 p1 + v2) (P1 p1 + v2 - P2 p2 - v3) (P2 p2 + v3 - v2 - P3 p3 v4) (v2 + P3 p3 + v4 - v3 - P4 p4) (v3 + P4 p4 - v4 - S1 s1 S2 s2 - v5) (v4 + S1 s1 + S2 s2 + v5 - P5 p5 - v6) (P5 p5 + v6 v5 - P6 p6) (v5 + P6 p6 - v6 - P7 p7) (v6 + P7 p7) (* pendant edges left to right - ^P# includes factor if exists *) (v2 + P1 p1)^P1 (v3 + P2 p2 - v2)^P2 (v2 + P3 p3 + v4 - v3)^ P3 (v3 + P4 p4 - v4)^ P4 (v4 + S1 s1 + S2 s2 + v5)^(S1 + S2) (P5 p5 + v6 - v5)^ P5 (v5 + P6 p6 - v6)^P6 (v6 + P7 p7)^P7 (* 3rd thread edge from stable vertex *) (v4 + S1 s1 + S2 s2 + v5) ); (* track the desired term in polynomial for Nullstellensatz *) coef[[n + 1]] = Coefficient[ f[v2, v3, v4, v5, v6, p1, p2, p3, p4, p5, p6, p7, s1, s2], v2^2 v3^2 v4^2 v5^2 v6^2 p1^P1 p2^P2 p3^P3 p4^P4 p5^P5 p6^P6 p7^ P7 s1^S1 s2^S2 ]; (* check for success *) If[coef[[n + 1]] == 0, Print["Failure with n=", n]; Break, If[n == 2^9 - 1, Print["Success!"] (* display coefficients *) Print[coef], Continue ] ] , {n, 0, 2^9 - 1}] Mathematica code for Lemma 3.5
(* 4-2-3 is 3-reducible - cycle through all possible combinations *) coef = Table[0, {i, 2^13}]; Do[ (* 13 independent choices for pendants - 9 p, 4 s *) {P1, P2, P3, P4, P5, P6, P7, P8, P9, S1, S2, S3, S4} = IntegerDigits[n, 2, 13]; (* indicator for pendants by name *) (* function obtained by deleting v2-v8 and any pendants *) f[v2_, v3_, v4_, v5_, v6_, v7_, v8_, p1_, p2_, p3_, p4_, p5_, p6_, p7_, p8_, p9_, s1_, s2_, s3_, s4_] := ( (* internal edges left to right - P# p# includes pendant if exists *) (P1 p1 + v2) (P1 p1 + v2 - P2 p2 - v3) (P2 p2 + v3 - v2 - P3 p3 v4) (v2 + P3 p3 + v4 - v3 - P4 p4) (v3 + P4 p4 - v4 - S1 s1 S2 s2 - v5) (v4 + S1 s1 + S2 s2 + v5 - P5 p5 - v6) (P5 p5 + v6 v5 - P6 p6) (v5 + P6 p6 - v6 - P3 p3 - P4 p4 - v7) (v6 + P3 p3 + P4 p4 + v7 - P7 p7 - v8) (P7 p7 + v8 - v7 - P8 p8) (v7 + P8 p8 - v8 - P9 p9) (v8 + P9 p9) (* pendant edges left to right - ^P# includes factor if exists *) (P1 p1 + v2)^P1 (P2 p2 + v3 - v2)^P2 (v2 + P3 p3 + v4 - v3)^ P3 (v3 + P4 p4 - v4)^ P4 (v4 + S1 s1 + S2 s2 + v5)^(S1 + S2) (P5 p5 + v6 - v5)^ P5 (v5 + P6 p6 - v6)^ P6 (v6 + S3 s3 + S4 s4 + v7)^(S3 + S4) (P7 p7 + v8 - v7)^ P7 (v7 + P8 p8 - v8)^P8 (v8 + P9 p9)^P9 (* 3rd thread edge from stable vertices *) (v4 + S1 s1 + S2 s2 + v5) (v6 + S3 s3 + S4 s4 + v7) ); (* track the desired term in polynomial for Nullstellensatz *) coef[[n + 1]] = Coefficient[ f[v2, v3, v4, v5, v6, v7, v8, p1, p2, p3, p4, p5, p6, p7, p8, p9, s1, s2, s3, s4], v2^2 v3^2 v4^2 v5^2 v6^2 v7^2 v8^2 p1^P1 p2^P2 p3^P3 p4^P4 p5^ Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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P5 p6^P6 p7^P7 p8^P8 p9^P9 s1^S1 s2^S2 s3^S3 s4^S4]; (* check for success *) If[coef[[n + 1]] == 0, Print["Failure with n=", n]; Break, If[n == 2^13 - 1, Print["Success!"](* display coefficients *) Print[coef], Continue ] ] , {n, 0, 2^13 - 1}]
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Planar graphs with girth 20 are additively 3-choosable ∗
Axel Brandt a , , Nathan Tenpas b , Carl R. Yerger c a
Northern Kentucky University, Highland Heights, KY 41099, United States of America Vanderbilt University, Nashville, TN 37235, United States of America c Davidson College, Davidson, NC 28035, United States of America b
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Article history: Received 17 May 2018 Received in revised form 20 June 2019 Accepted 28 August 2019 Available online xxxx Keywords: Additive coloring Lucky labeling Reducible configurations Discharging method Combinatorial Nullstellensatz
a b s t r a c t The additive choice number of a graph G, denoted by chΣ (G), is the minimum positive integer k such that, whenever each vertex is given a list of at least k positive integers, vertex labels can be chosen from respective lists in such a way that adjacent vertices have distinct sums of labels on their neighbors. Recently, bounds on the additive choice number have been proven for planar graphs under certain girth assumptions. In this paper, we prove that chΣ (G) ≤ 3 for the planar graph G with girth at least 20. Our approach uses the Combinatorial Nullstellensatz to streamline arguments within a proof using the discharging method. © 2019 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we consider only finite, undirected graphs. We abuse notation slightly by assuming a fixed plane embedding of a planar graph. When not otherwise specified, see [11] for notation. An additive coloring of a graph G is a labeling of the vertices of G with natural numbers such that two adjacent vertices have distinct sums of labels on their neighbors. The additive coloring number of G, denoted by χΣ (G), is defined to be the minimum natural number k for which G has an additive coloring using labels in {1, . . . , k}. We briefly mention that additive coloring was introduced in the literature as lucky labeling by Czerwiński, Grytczuk, and Żelazny [10]. It has also been referred to as open distinguishing by Axenovich et al. [4]. We direct the interested reader to [8] for a more detailed exposition of results in this area. In 2009, Czerwiński, Grytczuk, and Żelazny conjectured the following: Conjecture 1.1 ([10]). For every graph G, χΣ (G) ≤ χ (G). This conjecture is open in general, and χΣ (Kn ) = χ (Kn ) implies that the conjecture would be best possible if true. Of particular interest, there is currently no known constant bound for the class of bipartite graphs. The best known bound for the class of planar graphs is given by the following: Theorem 1.2 ([5]). If G is a planar graph, then χΣ (G) ≤ 468. Recently, significant progress has been made for planar graphs under certain girth assumptions. The girth of a graph G, denoted by girth(G), is the length of the shortest cycle in G. Bartnicki et al. [5] use the existence of an I,F-partition, which we do not discuss here, to show that χΣ (G) ≤ 4 for planar graphs G with girth at least 13. Using an improved result on the existence of I,F-partitions [6], their result was later improved as follows: ∗ Corresponding author. E-mail addresses: [email protected] (A. Brandt), [email protected] (N. Tenpas), [email protected] (C.R. Yerger). https://doi.org/10.1016/j.dam.2019.08.021 0166-218X/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Fig. 1. Configurations that do not appear in G: P(1, 0, 1), P(1, 1, 1), P(1, 1, 0, 0), P(0, 1, 0, 0), P(1, 0, 0, 0), and P(0, 0, 0, 0, 0).
Theorem 1.3 ([8]). If G is a planar graph with girth at least 10, then χΣ (G) ≤ 4. A list version of additive coloring has been studied, and similar results have been obtained. A graph G is additively k-choosable if an additive coloring can be selected from any list assignment of natural numbers to the vertices of G where each list has at least k distinct elements. The additive choice number of G, denoted by chΣ (G), is the minimum natural number k such that G is additively k-choosable. Ahadi and Dehghan [1] have shown that χΣ (G) and chΣ (G) can be arbitrarily far apart. Theorem 1.4 ([2]). If G is a forest, then chΣ (G) ≤ 3. Theorem 1.5 ([8]). Let G be a planar graph with girth g. If g ≥ 5, then chΣ (G) ≤ 19. If g ≥ 6, then chΣ (G) ≤ 9. If g ≥ 7, then chΣ (G) ≤ 8. If g ≥ 26, then chΣ (G) ≤ 3. In [7] the authors provide a shorter proof for the girth 26 result of Theorem 1.5. Expanding on that approach and incorporating aspects of the original proof enables the result to be strengthened as follows: Theorem 1.6.
If G is a planar graph with girth at least 20, then chΣ (G) ≤ 3.
2. Notation and tools As standard notation for proofs utilizing the discharging method, a j-, j− -, or j+ -vertex is a vertex with degree j, at most j, or at least j, respectively. Similarly, a j-, j− -, or j+ -neighbor of v is a j-, j− -, or j+ -vertex, respectively, adjacent to v . For the remainder of this paper, suppose G is a vertex-minimum counterexample to Theorem 1.6. That is, let G be a planar graph with girth at least 20 and chΣ (G) ≥ 4 such that chΣ (H) ≤ 3 for any graph H with |V (H)| < |V (G)|. By the nature of additive coloring, G is connected. To show that G does not exist, we utilize the following from [8] as it applies to our main result, that is when k = 3 and Q = ∅: Lemma 2.1 ([8], Lemma 3.1 Part b). A vertex v in G with r 1-neighbors satisfies 1 + 2r ≤ d(v ). For v ∈ V (G) and S ⊆ V (G), let dist(v, S) be the minimum distance from v to some vertex in S. As such, for a vertex v , define the cycle distance of v , denoted by cdist(v ), to be the minimum distance from v to a cycle of G, that is cdist(v ) := min{dist(v, V (C )): C is a cycle in G}. Note that this definition is well-defined because Theorem 1.4 implies G contains a cycle. The following appears in [7], and we include the proof for completeness. Lemma 2.2 ([7]). Let v be a 1-vertex in G. Then cdist(v ) = 1. Proof. Assume cdist(v ) ≥ cdist(u) for every 1-vertex u in G. Toward a contradiction, suppose cdist(v ) ≥ 2. Let w be the neighbor of v . By the choice of v , d(w ) ≥ 2 and w has exactly one 2+ -neighbor. Hence, w has (d(w ) − 1) 1-neighbors. Since 1 + 2(d(w ) − 1) = 2d(w ) − 1 ≥ d(w ) + 1 > d(w ), Lemma 2.1 implies that w does not exist in G, a contradiction. □ The configurations in Fig. 1 do not appear in G by Lemma 3.8 in [8]. Toward describing these and similar configurations, let P(t2 , . . . , tn−1 ) be the path v1 · · · vn such that, for each i in {2, . . . , n − 1}, the vertex vi has ti 1-neighbors and d(vi ) = 2 + ti . Note that Lemma 2.1 implies that ti ≤ 1 for each such vi . The following appears in [8] without proof, which we provide here for completeness. Lemma 2.3 ([8]). Any P(t2 , . . . , tn−1 ) in G satisfies n ≤ 6. Proof. The following table lists all possibilities for P(t2 , . . . , t6 ) along with the configurations pictured in Fig. 1 that preclude their existence in G. Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Fig. 2. An image of P(1, 1, 0) ⊙ P(1, 0, 0, 1) with respective underlying paths v1 · · · v5 and v1′ · · · v6′ .
Fig. 3. A visual depiction of G where the ellipses represent the rest of G, each of which includes a cycle containing u, x, y, and z, respectively, and are only connected by paths containing at least three labeled vertices.
Configuration
Forbids P(t2 , . . . , t6 )
P(1, 0, 1)
00101, 10110, 00111, 00011, 00010, 00001, 00000
P(1, 1, 1) P(1, 1, 0, 0) P(0, 1, 0, 0) P(1, 0, 0, 0) P(0, 0, 0, 0, 0)
01010, 10111, 01110, 00110, 00100, 10000,
01011, 11010, 01111, 01100, 01000, 10001
01101, 11011, 11100, 10011, 01001,
10100, 10101, 11101 11110, 11111 11000, 11001 10010
This list suffices because any P(t2 , . . . , tn−1 ) with n > 6 would necessarily contain one of the P(t2 , . . . , t6 ) listed above.
□
For P(t2 , . . . , tm−1 ) with underlying path v1 · · · vm , let P(t2 , . . . , tn−1 ) ⊙ P(t2 , . . . , tm−1 ) denote the graph obtained from ′ ′ P(t2 , . . . , tn−1 ) and P(t2′ , . . . , tm −1 ) by identifying vn and v1 . See Fig. 2 for an example of this construction. As in [7], define the thread degree of a vertex v , denoted by d2+ (v ), as the number of 2+ -neighbors of v . A hub vertex is a vertex v with d2+ (v ) ≥ 3, and a stable vertex is a hub vertex contained in a cycle. Define a (u, v )-thread to be a path whose endpoints u and v are hub vertices, and whose internal vertices are 2-vertices. In Fig. 3, all labeled vertices are hubs, but v and w are not stable. Note that Lemma 2.2 implies that the set of 2-vertices of G is a subset of the set of internal vertices of (u, v )-threads in G. Also, every 1-vertex of G is either adjacent to a hub vertex, or in some P(t2 , . . . , tn−1 ). ′
′
′
′
′
′
3. Some additional structure In this section we describe additional structural properties of G. Specifically, we enumerate some configurations that do not appear in G. We will utilize the following result of Alon: Theorem 3.1 field F. If there is a ∏ (Combinatorial ∑ Nullstellensatz [3]). Let f be a polynomial of degree t in m variables over a ∏ t monomial xi i in f with ti = t whose coefficient is nonzero in F, then f is nonzero at some point of Ti , where each Ti is a set of ti + 1 distinct values in F. We follow the approach of [8], which builds a polynomial whose nonzero solutions satisfy the restrictions for additive coloring. We include the following as an example. Lemma 3.2.
The configurations P(1, 1, 0) and P(1, 0, 0, 1) do not appear in G.
Proof. Let L be a function that assigns every vertex of G a list of three natural numbers. Suppose that P(1, 1, 0) appears in G. Let v1 · · · v5 be the underlying path of P(1, 1, 0) and let p2 and p3 be the pendant vertices adjacent to v2 and v3 , respectively. Let G′ = G − {p2 , p3 , v3 }. By the minimality of G, chΣ (G′ ) ≤ 3. As such, let ℓ be an additive coloring from L on V (G′ ). We will use Theorem 3.1 to guarantee that there exist ℓ(p2 ) ∈ L(p2 ), ℓ(p3 ) ∈ L(p3 ), and ℓ(v3 ) ∈ L(v3 ) that extend ℓ to be an additive coloring of G. Notice that the only edges impacted by the labels of p2 , p3 , and v3 are vi vi+1 for i ∈ {1, 2, 3, 4}, v2 p2 , and v3 p3 . The following polynomial has variables x, y, and z corresponding to the choice of labels at p2 , p3 , and v3 , respectively, and Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Fig. 4. A picture of P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ , t4′ ), where dash-dot edges indicate the potential of that edge’s existence.
has factors corresponding to a difference of label sums at endpoints of the edges v1 v2 , v2 p2 , v2 v3 , v3 p3 , v3 v4 , and v4 v5 , respectively: f (x, y, z) = (x + z + ℓ(v1 ) − S(v1 )) × (ℓ(v1 ) + x + z − ℓ(v2 ))
× (ℓ(v1 ) + x + z − ℓ(v2 ) − y − ℓ(v4 )) × (ℓ(v2 ) + y + ℓ(v4 ) − z) × (ℓ(v2 ) + y + ℓ(v4 ) − z − ℓ(v5 )) × (z + ℓ(v5 ) − S(v5 )), where S(v ) = u∈N(v ) ℓ(u). Notice that each factor of f is linear with respect to some variable. Thus, selecting a constant from a factor when expanding f causes the degree of the monomial obtained to be less than the degree of f . As such, the coefficient of x2 y2 z 2 in f (x, y, z) is equal to its coefficient in (x + z)2 (x + z − y)(y − z)2 z, which is 5. By Theorem 3.1, there is a choice of ℓ(p2 ) ∈ L(p2 ), ℓ(p3 ) ∈ L(p3 ), and ℓ(v3 ) ∈ L(v3 ) that is a nonzero solution of f . These labels then extend ℓ to an additive coloring of G. Hence, chΣ (G) ≤ 3, a contradiction with the choice of G. The proof for P(1, 0, 0, 1) is similar. Let v1 · · · v6 be the underlying path of P(1, 0, 0, 1) and let p2 and p5 be the pendant vertices adjacent to v2 and v5 , respectively. Let G′ = G − {p2 , v3 , v4 , p5 }. The following polynomial has variables w, x, y, and z corresponding to the choice of labels at p2 , v3 , v4 , and p5 and factors corresponding to the edges v1 v2 , v2 p2 , v2 v3 , v3 v4 , v4 v5 , v5 p5 , and v5 v6 , respectively:
∑
g(w, x, y, z) = (ℓ(v1 ) + w + x − S(v1 )) × (ℓ(v1 ) + w + x − ℓ(v2 ))
× (ℓ(v1 ) + w + x − ℓ(v2 ) − y) × (ℓ(v2 ) + y − x − ℓ(v5 )) × (x + ℓ(v5 ) − y − z − ℓ(v6 )) × (y + z + ℓ(v6 ) − ℓ(v5 )) × (y + z + ℓ(v6 ) − S(v6 )), where S(v ) = u∈N(v ) ℓ(u). As before, we may omit constants without altering the coefficient of maximum degree terms. The coefficient of w 2 x2 y2 z in g(w, x, y, z) is equal to its coefficient in (w + x)2 (w + x − y)(y − x)(x − y − z)(y + z)2 , which is 17. □
∑
The following lemmas are obtained using Mathematica to loop through all possible configurations. See the Appendix for relevant code. For an intuition into coding polynomials that allow for the possible nonexistence of edges, we use the configuration in Fig. 4 as an example. For this example, we will abuse notation by using vi , pi , vi as both names of vertices and the names for corresponding variables in the polynomial. We delete the labeled vertices (where each pi may or may not initially exist). Let Pi and Si be {0, 1}-indicators for the existence of each pi and si , respectively. Our code loops through all 27+2 possible indicator assignments, which cover all potential subconfigurations. These indicator assignments are used as exponents on factors and coefficients on variables in sums. For example, the factor corresponding to the edge v3 p3 is (v2 + P3 p3 + v4 − v3 )P3 . Toward verifying the polynomials cover all possibilities, note that Lemma 2.1 implies that vertices with thread degree 3 have at most two 1-neighbors. Lemma 3.3. Let v1 · · · v13 be the underlying path of P(t2 , t3 , t4 ) ⊙ P(t2′ , t3′ , t4′ ) ⊙ P(t2′′ , t3′′ , t4′′ ). If d2+ (v5 ) = d2+ (v9 ) = 3, then P(t2 , t3 , t4 ) ⊙ P(t2′ , t3′ , t4′ ) ⊙ P(t2′′ , t3′′ , t4′′ ) does not appear in G. Lemma 3.4. Let v1 · · · v10 be the underlying path of P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ , t4′ ). If d2+ (v6 ) = 3, then P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ , t4′ ) does not appear in G. Lemma 3.5. Let v1 · · · v13 be the underlying path of P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ ) ⊙ P(t2′′ , t3′′ , t4′′ ). If d2+ (v6 ) = d2+ (v9 ) = 3, then P(t2 , t3 , t4 , t5 ) ⊙ P(t2′ , t3′ ) ⊙ P(t2′′ , t3′′ , t4′′ ) does not appear in G. 4. Proof of Theorem 1.6 We construct a graph G′ from G and use a discharging argument to show that G′ , and thus G, does not exist. Our construction of G′ from G follows these steps in order: Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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(S1) Delete all 1-vertices of G. (S2) For all (u, v )-threads with a cut-edge, contract all edges of the (u, v )-thread. (S3) Replace all remaining (u, v )-threads with the edge uv . Recall that G is simple, planar, connected, and has girth at least 20. Using Lemma 2.3, it is straightforward to show that G′ is simple, planar, and connected with δ (G′ ) ≥ 3 and girth(G′ ) ≥ 4. Thus G′ is also 2-edge-connected. Proposition 3.1 in [9] gives
∑
(d(v ) − 4) +
v∈V (G′ )
∑
(l(f ) − 4) = −8.
f ∈F (G′ )
Thus assigning each vertex v an initial charge µ(v ) = d(v ) − 4 and each face f an initial charge µ(f ) = l(f ) − 4 gives an initial total charge of −8. Consider distributing charge from faces to vertices as follows: (R1) Each 3-vertex receives
1 3
charge from each of its incident faces.
We will show that distributing charge in this way results in every vertex and face of G′ having a nonnegative final charge, for a total charge at least 0, which is an impossibility. This impossibility implies that G′ does not exist, and, by construction, that G does not exist. Hence, no counterexample exists to Theorem 1.6. Every Vertex of G′ has Nonnegative Final Charge We first consider the final charge of a vertex v . If d(v ) ≥ 4, then v does not lose or gain any charge under (R1). Thus v would have final charge µ(v ) = d(v ) − 4 ≥ 0. Since δ (G′ ) ≥ 3, it remains to consider when d(v ) = 3. In this case, the 2-edge-connectivity of G′ implies that v is incident to 3 faces. Thus v has final charge µ(v ) + 3( 31 ) = (3 − 4) + 1 = 0. Therefore, all vertices of G′ have nonnegative final charge. Every Face of G′ has Nonnegative Final Charge We now turn our attention to the faces of G′ . Let f be a face of G′ . If f is incident to t 3-vertices, then f has final charge l(f ) − 4 − t( 31 ) ≥ 23 l(f ) − 4. Thus if l(f ) ≥ 6, then f has nonnegative final charge. Since girth(G′ ) ≥ 4, it remains to consider when l(f ) ∈ {4, 5}. From the previous calculation, it suffices to show that f is incident to four 4+ -vertices when l(f ) = 4 and to show that f is incident to at least two 4+ -vertices when l(f ) = 5. In constructing G′ from G, (S3) replaces each (u, v )-thread with an edge. We say that (E1 , E2 , . . . , El(f ) ) is a replacement sequence for a face f if the edges incident to f , listed consecutively, were obtained during (S3) by replacing Ei vertices in a (u, v )-thread. Let l(f ) = 4. Since girth(G) ≥ 20, there must be at least 16 internal vertices of (u, v )-threads replaced during (S3). Lemma 2.3 implies that each edge in the boundary of f was obtained during (S3) from a (u, v )-thread with at most 4 internal vertices. Thus f is a (4, 4, 4, 4)-face. By Lemma 3.4, all four vertices incident to f are 4+ -vertices, as desired. Let l(f ) = 5. Since girth(G) ≥ 20, there must be at least 15 internal vertices of (u, v )-threads replaced during (S3). Lemma 2.3 implies that there are at most 20 internal vertices of (u, v )-threads replaced during (S3). We consider all of the ways to distribute these vertices to edges in the boundary of f , and will enumerate the cases based on the number 4s in the replacement sequence of f . Let the replacement sequence of f have no 4. Then f is a (3, 3, 3, 3, 3) face. Two applications of Lemma 3.3 imply that f is incident at least two 4+ -vertices, as desired. Let the replacement sequence of f have exactly one 4. Since there are either 11 or 12 vertices replaced by the other edges of f , the replacement sequence has at least three 3s and at most one 2. Thus the replacement sequence of f , up to symmetry, is (4, 3, 3, 3, 3), (4, 3, 3, 2, 3), or (4, 3, 3, 3, 2). For the first two possible replacement sequences of f , two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. For the last possible replacement sequence of f , applications of Lemmas 3.3 and 3.4 imply that f is incident to at least two 4+ -vertices, as desired. Let the replacement sequence of f have exactly two 4s. First, assume that the two 4s are consecutive. If (4, 4, 3) is a subsequence of the replacement sequence of f , then two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. Otherwise, the replacement sequence of f is (4, 4, 2, 3, 2) and applications of Lemmas 3.4 and 3.5 imply that f is incident to at least two 4+ -vertices. Now, assume that the two 4s are not consecutive. If either (4, 3, 4) or (4, 3, 3, 4) is a subsequence of the replacement sequence of f , then two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. Otherwise, (4, 2, 4, 3) is a subsequence of the replacement sequence of f and applications of Lemmas 3.4 and 3.5 imply that f is incident to at least two 4+ -vertices, as desired. Finally, let the replacement sequence of f have at least three 4s. If (4, 4, 4) is a subsequence of the replacement sequence of f , then two applications of Lemma 3.4 imply that f is incident to at least two 4+ -vertices. Otherwise, (4, 4, 3, 4) or (4, 4, 2, 4) is a subsequence of the replacement sequence of f , and applications of Lemma 3.4, and Lemma 3.4 or 3.5, respectively, imply that f is incident to at least two 4+ -vertices, as desired. Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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Appendix
Mathematica code for Lemma 3.3
(* 3-3-3 is 3-reducible - cycle through all possible combinations *) coef = Table[0, {i, 2^13}]; Do[ (* 13 independent choices for pendants - 9 p, 4 s *) {P1, P2, P3, P4, P5, P6, P7, P8, P9, S1, S2, S3, S4} = IntegerDigits[n, 2, 13]; (* 0-1 indicator for pendants by name *) (* function obtained by deleting v2-v8 and any pendants that exist *) f[v2_, v3_, v4_, v5_, v6_, v7_, v8_, p1_, p2_, p3_, p4_, p5_, p6_, p7_, p8_, p9_, s1_, s2_, s3_, s4_] := ( (* internal edges left to right - P# p# includes pendant if exists *) (P1 p1 + v2) (P1 p2 + v2 - P2 p2 - v3) (P2 p2 + v3 - v2 P3 p3) (v2 + P3 p3 - v3 - S1 s1 - S2 s2 - v4) (v3 + S1 s1 + S2 s2 + v4 - P4 p4 - v5) (P4 p4 + v5 - v4 - P5 p5 - v6) (v4 + P5 p5 + v6 - v5 - P6 p6) (v5 + P6 p6 - v6 - S3 s3 - S4 s4 v7) (v6 + S3 s3 + S4 s4 + v7 - P7 p7 - v8) (P7 p7 + v8 - v7 P8 p8) (v7 + P8 p8 - v8 - P9 p9) (v8 + P9 p9) (* pendant edges left to right - ^P# includes factor if edge exists *) (P1 p1 + v2)^P1 (P2 p2 + v3 - v2)^P2 (v2 + P3 p3 - v3)^ P3 (v3 + S1 s1 + S2 s2 + v4)^(S1 + S2) (P4 p4 + v5 - v4)^ P4 (v4 + P5 p5 + v6 - v5)^P5 (v5 + P6 p6 - v6)^ P6 (v6 + S3 s3 + S4 s4 + v7)^(S3 + S4) (P7 p7 + v8 - v7)^ P7 (v7 + P8 p8 - v8)^P8 (v8 + P9 p9)^P9 (* 3rd thread edge from stable vertices *) (v3 + S1 s1 + S2 s2 + v4) (v6 + S3 s3 + S4 s4 + v7) ); (* track the desired term in polynomial for Nullstellensatz *) coef[[n + 1]] = Coefficient[ f[v2, v3, v4, v5, v6, v7, v8, p1, p2, p3, p4, p5, p6, p7, p8, p9, s1, s2, s3, s4], (* base vertices *) v2^2 v3^2 v4^2 v5^2 v6^2 v7^2 v8^2 (* pendants *) p1^P1 p2^P2 p3^P3 p4^P4 p5^P5 p6^P6 p7^P7 p8^P8 p9^P9 (* stable vtx *) s1^S1 s2^S2 s3^S3 s4^S4]; (* check for success *) If[coef[[n + 1]] == 0, Print["Failure with n=", n]; Break, If[n == 2^13 - 1, Print["Success!"] (* Display coefficients *) Print[coef], Continue ] ] , {n, 0, 2^13 - 1}] Mathematica code for Lemma 3.4
(* 4-3 is 3-reducible - cycle through all possible combinations *) coef = Table[0, {i, 2^9}]; Do[ (* 9 independent choices for pendants - 7 p, 2 s *) {P1, P2, P3, P4, P5, P6, P7, S1, S2} = IntegerDigits[n, 2, 9]; (* indicator for pendants by name *) (* function obtained by deleting v2-v8 and any pendants *) f[v2_, v3_, v4_, v5_, v6_, p1_, p2_, p3_, p4_, p5_, p6_, p7_, s1_, s2_] := ( (* internal edges left to right - P# p# includes pendant if exists *) Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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(P1 p1 + v2) (P1 p1 + v2 - P2 p2 - v3) (P2 p2 + v3 - v2 - P3 p3 v4) (v2 + P3 p3 + v4 - v3 - P4 p4) (v3 + P4 p4 - v4 - S1 s1 S2 s2 - v5) (v4 + S1 s1 + S2 s2 + v5 - P5 p5 - v6) (P5 p5 + v6 v5 - P6 p6) (v5 + P6 p6 - v6 - P7 p7) (v6 + P7 p7) (* pendant edges left to right - ^P# includes factor if exists *) (v2 + P1 p1)^P1 (v3 + P2 p2 - v2)^P2 (v2 + P3 p3 + v4 - v3)^ P3 (v3 + P4 p4 - v4)^ P4 (v4 + S1 s1 + S2 s2 + v5)^(S1 + S2) (P5 p5 + v6 - v5)^ P5 (v5 + P6 p6 - v6)^P6 (v6 + P7 p7)^P7 (* 3rd thread edge from stable vertex *) (v4 + S1 s1 + S2 s2 + v5) ); (* track the desired term in polynomial for Nullstellensatz *) coef[[n + 1]] = Coefficient[ f[v2, v3, v4, v5, v6, p1, p2, p3, p4, p5, p6, p7, s1, s2], v2^2 v3^2 v4^2 v5^2 v6^2 p1^P1 p2^P2 p3^P3 p4^P4 p5^P5 p6^P6 p7^ P7 s1^S1 s2^S2 ]; (* check for success *) If[coef[[n + 1]] == 0, Print["Failure with n=", n]; Break, If[n == 2^9 - 1, Print["Success!"] (* display coefficients *) Print[coef], Continue ] ] , {n, 0, 2^9 - 1}] Mathematica code for Lemma 3.5
(* 4-2-3 is 3-reducible - cycle through all possible combinations *) coef = Table[0, {i, 2^13}]; Do[ (* 13 independent choices for pendants - 9 p, 4 s *) {P1, P2, P3, P4, P5, P6, P7, P8, P9, S1, S2, S3, S4} = IntegerDigits[n, 2, 13]; (* indicator for pendants by name *) (* function obtained by deleting v2-v8 and any pendants *) f[v2_, v3_, v4_, v5_, v6_, v7_, v8_, p1_, p2_, p3_, p4_, p5_, p6_, p7_, p8_, p9_, s1_, s2_, s3_, s4_] := ( (* internal edges left to right - P# p# includes pendant if exists *) (P1 p1 + v2) (P1 p1 + v2 - P2 p2 - v3) (P2 p2 + v3 - v2 - P3 p3 v4) (v2 + P3 p3 + v4 - v3 - P4 p4) (v3 + P4 p4 - v4 - S1 s1 S2 s2 - v5) (v4 + S1 s1 + S2 s2 + v5 - P5 p5 - v6) (P5 p5 + v6 v5 - P6 p6) (v5 + P6 p6 - v6 - P3 p3 - P4 p4 - v7) (v6 + P3 p3 + P4 p4 + v7 - P7 p7 - v8) (P7 p7 + v8 - v7 - P8 p8) (v7 + P8 p8 - v8 - P9 p9) (v8 + P9 p9) (* pendant edges left to right - ^P# includes factor if exists *) (P1 p1 + v2)^P1 (P2 p2 + v3 - v2)^P2 (v2 + P3 p3 + v4 - v3)^ P3 (v3 + P4 p4 - v4)^ P4 (v4 + S1 s1 + S2 s2 + v5)^(S1 + S2) (P5 p5 + v6 - v5)^ P5 (v5 + P6 p6 - v6)^ P6 (v6 + S3 s3 + S4 s4 + v7)^(S3 + S4) (P7 p7 + v8 - v7)^ P7 (v7 + P8 p8 - v8)^P8 (v8 + P9 p9)^P9 (* 3rd thread edge from stable vertices *) (v4 + S1 s1 + S2 s2 + v5) (v6 + S3 s3 + S4 s4 + v7) ); (* track the desired term in polynomial for Nullstellensatz *) coef[[n + 1]] = Coefficient[ f[v2, v3, v4, v5, v6, v7, v8, p1, p2, p3, p4, p5, p6, p7, p8, p9, s1, s2, s3, s4], v2^2 v3^2 v4^2 v5^2 v6^2 v7^2 v8^2 p1^P1 p2^P2 p3^P3 p4^P4 p5^ Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.
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P5 p6^P6 p7^P7 p8^P8 p9^P9 s1^S1 s2^S2 s3^S3 s4^S4]; (* check for success *) If[coef[[n + 1]] == 0, Print["Failure with n=", n]; Break, If[n == 2^13 - 1, Print["Success!"](* display coefficients *) Print[coef], Continue ] ] , {n, 0, 2^13 - 1}]
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Please cite this article as: A. Brandt, N. Tenpas and C.R. Yerger, Planar graphs with girth 20 are additively 3-choosable, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.08.021.