Complexity of 3-Edge-Coloring in the Class of Cubic Graphs with a Polyhedral Embedding in an Orientable Surface

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Discrete Applied Mathematics 158 (2010) 1856–1860

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Discrete Applied Mathematics

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Note Complexity of 3-edge-coloring in the class of cubic graphs with a polyhedral embedding in an orientable surface$ Martin Kochol MÚ SAV, Štefánikova 49, 814 73 Bratislava 1, Slovakia article info a b s t r a c t

Article history: A polyhedral embedding in a surface is one in which any two faces have boundaries that Received 3 September 2009 are either disjoint or simply connected. In a cubic (3-regular) graph this is equivalent to Received in revised form 23 June 2010 the dual being a simple graph. In 1968, Grünbaum conjectured that every cubic graph Accepted 28 June 2010 with a polyhedral embedding in an orientable surface is 3-edge-colorable. For the sphere, Available online 22 July 2010 this is equivalent to the Four-Color Theorem, but we have disproved the conjecture in the general form. In this paper we extend this result and show that if we restrict our attention Keywords: to a class of cubic graphs with a polyhedral embedding in an orientable surface, then the Edge-coloring Orientable surface computational complexity of the 3-edge-coloring problem and its approximation does not Polyhedral embedding improve. NP-complete problem © 2010 Elsevier B.V. All rights reserved. Approximation algorithms Cyclical edge-connectivity

1. Introduction

A k-edge-coloring of a graph is an assignment of numbers 1,..., k to its edges such that any two incident edges receive different colors. By Tait [17], a cubic (3-regular) planar graph is 3-edge-colorable if and only if its geometric dual is 4-colorable. Thus the dual form of the Four-Color Theorem (see [1]) is that every 2-edge-connected planar cubic graph has a 3-edge-coloring. Denote by C the class of cubic graphs. From the classical result of Vizing [19], it follows that there is a polynomial time algorithm which finds a 4-edge-coloring for any graph of C (see also [14]). On the other hand, by Holyer [8], the problem of deciding whether a graph from C is 3-edge-colorable is NP-complete. For G ∈ C, define by σ (G) the minimum number such that there exists a 4-edge-coloring of G with σ (G) edges assigned the fourth color. Define by ρ(G) the minimum number of vertices that must be deleted from G to have the resulting graph be 3-edge-colorable. Clearly, σ (G) = ρ(G) = 0 if and only if G is 3-edge-colorable. In [15] it was proved that the problems of deciding whether ρ(G), σ (G) ∈ [0, n1−] are NP-complete for G ∈ C. Thus, it is an NP-hard problem to approximate ρ(G) and σ (G), G ∈ C, with an error O(n1−). An embedding of a cubic graph in a surface is called polyhedral if it is cellular (each face is homeomorphic to an open disk) and its dual is a simple graph (i.e., any two faces have at most one edge in common and the boundary of each face is a simple circuit; see, e.g., [2]). Let P denote the class of cubic graphs with a polyhedral embedding in an orientable surface (see, e.g., [3,6] for a formal definition of orientable surfaces). During a conference in 1968, Grünbaum [7] presented a conjecture that every graph from P is 3-edge-colorable. By the results of Appel and Haken [1] and Tait [17], the conjecture holds true for the sphere. A positive solution of this conjecture would generalize the dual form of the Four-Color Theorem to every orientable surface. We disproved the general form of the conjecture in [9,10].

$ This work was partially supported by grant VEGA 2/0118/10. E-mail addresses: [email protected], [email protected].

Fig. 1.

In this paper we continue with the study begun in [9,10] and show that 3-edge-coloring problems have the same computational complexity on classes P and C. We generalize the results from [8,15] and prove that the problem of deciding whether a graph from P is 3-edge-colorable is NP-complete and the problems of deciding whether ρ(G), σ (G) ∈ [0, n1−] are NP-complete for G ∈ P .

2. Networks, flows and superposition

If G is a graph, then V (G) and E(G) denote the vertex and edge sets of G, respectively. If v is a vertex of G, then ωG(v) denotes the set of edges having one end v and the other end from V (G) \{v}. By a network we mean a couple (G, U) where G is a graph and U ⊆ V (G). By a nowhere-zero Z2 × Z2-flow ϕ in (G, U) we mean a mapping ϕ : E(G) → × such that ϕ(e) 6= 0 for each edge e of G and ∂ϕ(v) = P ϕ(e) = 0 for each Z2 Z2 e∈ωG(v) vertex v ∈ V (G) \ U. By a nowhere-zero Z2 × Z2-flow in a graph G we mean a nowhere-zero Z2 × Z2-flow in (G, ∅). If G is a graph, then denote by ρ4(G) the minimum number n such that there exists U ⊆ V (G), |U| = n, such that (G, U) has a nowhere-zero Z2 × Z2-flow (see [11]). In [12], we proved the following statement.

Lemma 1. For every loopless cubic graph G, ρ4(G) = ρ(G) = σ (G). A graph is called cyclically k-edge-connected if deleting fewer than k edges does not result in a graph having at least two components containing cycles. Cubic graphs can have edge-connectivity at most 3, but arbitrarily large cyclic edge- connectivity. Denote by Ck and Pk the classes of graphs from C and P that are cyclically k-edge-connected, respectively. Nontrivial cubic graphs without a 3-edge-coloring are called snarks. By nontrivial we mean cyclically 4-edge-connected and with girth (length of the shortest circuit) at least 5. The best known snark is the Petersen graph. Snarks present an important class of graphs, because the smallest counterexamples to many conjectures about graphs must be among them (see, e.g., [11,13]). By a 4-snark we mean a network or a graph without a nowhere-zero Z2 × Z2-flow. It is well known that nowhere-zero Z2 × Z2-flows in a cubic graph G correspond to 3-edge-colorings of G by nonzero elements of the group Z2 × Z2 (see, e.g., [11,13]). Thus snarks form a proper subclass of 4-snarks. In [11,13] we introduced a general method for constructing 4-snarks. It is based on the following two steps. 0 Suppose v is a vertex of a graph G and G arises from G by the following process. Replace v by a graph Hv so that each edge e of G having one end v now has one end from Hv. If e is a loop and has both ends equal to v, then both ends of e will 0 now be from Hv. Thus constructed, G is called a vertex superposition of G. 0 Suppose e is an edge of G with ends u and v, and G arises from G by the following process. Replace e by a graph He having 0 0 0 0 0 at least two vertices, i.e., we delete e; pick two distinct vertices u , v of He; and identify u with u and v with v. Then G is 0 called an edge superposition of G. Furthermore, if He is a 4-snark, then G is called a 4-strong edge superposition of G. We say that a graph G0 is a (4-strong) superposition of G if G0 arises from G after finitely many vertex and (4-strong) edge superpositions. In [11, Lemma 4.4], we proved that a 4-strong superposition of a 4-snark is a 4-snark. We use a stronger form of this statement, proved in [11, Proposition 8.1, items (b) and (c)]. 0 0 Lemma 2. If G is a 4-strong superposition of G, then ρ4(G ) ≥ ρ4(G).

3. Reduction

In Fig. 1 there is depicted a snark G22 constructed in [9,10]. It is drawn in a plane with two handles. (A handle in a plane or a surface arises after deleting an open rectangle and identifying the opposite segments so that the orientations of the arrows, as indicated in Fig. 1, are preserved.) The boundary of the infinite face f0 is a circuit C, which is composed from two paths P1 and P2 with ends u and v. By [10, items (4) and (5)], we have:

(1) any two faces fi and fj, i, j ∈ {1,..., 8}, share at most one edge, (2) the face f0 shares exactly two edges with each fi, i ∈ {1,..., 8}, so P1 and P2 each contain exactly one of these edges.

In Fig. 2 is indicated a labeling of edges of G22 such that the edges incident with u or v have the same color and the edges incident with any other vertex have three different colors.

Lemma 3. Suppose G is a connected cubic graph of order n and let d ≥ 0 be an integer. Then we can construct in polynomial 2 time a graph Gd ∈ P5 of order 37 n + 2d such that ρ4(Gd) ≥ ρ4(G) and ρ4(Gd) = 0 if ρ4(G) = 0. 1858 M. Kochol / Discrete Applied Mathematics 158 (2010) 1856–1860

Fig. 2.

Fig. 3.

Fig. 4.

Proof. In the proof we assume familiarity with the basic concepts of topological graph theory as presented in [6,16]. Using the rotational scheme of a graph (see, e.g., [4,6,16,18]), we can construct a cellular embedding of G into an orientable surface in O(n) time. Note that its dual can have multiple edges and loops; thus it does not need to be polyhedral. Replace in G edges by copies of graph G22. All original vertices of G will have degree 9. Replace each of them by seven vertices of degree 3 as indicated in Fig. 3 (supposing the fragments of edges colored with one color correspond to the edges 0 incident with u or v in a copy of G22). Denote the resulting cubic graph by G . 0 0 Since all edges of G are replaced by snarks, G is a 4-proper superposition of G. Thus by Lemma 2, ρ4(G ) ≥ ρ4(G). Suppose G has a 3-edge-coloring. Then this indicates a 3-edge-coloring of G0 as follows: if an edge e of G has color i ∈ {1, 2, 3}, then label the edges of the copy of G22 replacing e, applying a permutation of the colors on Fig. 2, so that all six edges incident with u or v have color i, and color the rest of the edges applying suitable permutations of colors on Fig. 3. Therefore, if G is 3-edge-colorable then so is G0. G has n vertices and 3n/2 edges (n is even). Thus G0 has order 20(3n/2) + 7n = 37n. If we start with G embedded into a 0 surface of genus g, then we get an embedding of G into a surface of genus g + 3n (with each copy of G22 we add two new handles). The embedding is cellular, and by (1) and (2), its dual has no loops (but can have multiple edges). 0 2 Now repeating the same construction, we get from G a graph G0 of order 37 n, with a polyhedral embedding in an orientable surface of genus g + 3n + 3(37)n = g + 114n. Furthermore, G0 is cyclically 5-edge-connected, ρ4(G0) ≥ ρ4(G), and ρ4(G0) = 0 if ρ4(G) = 0. We can replace some vertices of G0 by a copy of the graph from Fig. 4. Furthermore, the 6-circuit of the graph from Fig. 4 can be recursively replaced as indicated in Fig. 5. Thus, vertices of G0 can be replaced by graphs having 7 or 9 + 12k, k ≥ 0, 0 2 vertices. Since n ≥ 2, G has at least 74 vertices. In this way we get, for each d ≥ 0, a graph Gd ∈ P5 of order 37 n + 2d such that ρ4(Gd) ≥ ρ4(G), and ρ4(Gd) = 0 if ρ4(G) = 0. This proves the statement.

4. Complexity of 3-edge-coloring problems

We suppose familiarity with terminology and results of NP-completeness as presented in [5].

Theorem 4. The problem to decide whether a graph G ∈ P5 has a 3-edge-coloring is NP-complete. M. Kochol / Discrete Applied Mathematics 158 (2010) 1856–1860 1859

Fig. 5.

Proof. Let G be a connected cubic graph of order n. By Lemma 3, we can construct in a polynomial time a graph G0 of order 2 37 n such that G0 is 3-edge-colorable if and only if G is. Thus there exists a polynomial transformation from the problem for 3-edge-coloring of connected cubic graphs, which is NP-complete by Holyer [8], to our problem.

Theorem 5. Let 0 < < 1 be a constant. Then the following problems are NP-complete: 1− 1− 1− Instance: A graph G ∈ P5 of order n. Question: Is ρ4(G) ∈ [0, n ] (or ρ(G) ∈ [0, n ] and σ (G) ∈ [0, n ])?

Proof. By Lemma 1, it suffices to prove the statement for ρ4. Let m > 0 be an even integer. Choose r = [372(m + 98)]d(1−)/e and n = 372(m + 98)r. We claim that r ≥ n1−. In fact, − r ≥ [372(m + 98)](1 )/, − r ≥ [372(m + 98)]1 , − − − r = rr1 ≥ [372(m + 98)r]1 = n1 . Let H be a connected cubic graph of order m. In the proof of [15, Theorem 2], a connected graph G of order (m + 98)r was constructed from H such that if ρ(H) = 0, then ρ(G) = 0 and if ρ(H) > 0, then ρ(G) ≥ r. Hence by Lemma 1, if ρ4(H) = 0, 2 then ρ4(G) = 0 and if ρ4(H) > 0, then ρ4(G) ≥ r. By Lemma 3, we can construct a graph G0 ∈ P5 of order 37 (m+98)r = n 1− such that ρ4(G0) ≥ ρ4(G) ≥ r ≥ n if ρ4(H) > 0 and ρ4(G0) = 0 if ρ4(G) = ρ4(H) = 0. By [15, Theorem 2], Lemma 3, and the choice of r and n, the construction runs in polynomial time. Thus there exists a polynomial transformation from the problem for 3-edge-coloring of connected cubic graphs, which is NP-complete by Holyer [8], to our problem.

1− Theorem 6. The problem of approximating ρ4(G), or ρ(G) and σ (G),G ∈ P5, with an error O(n ) (where n denotes the order of G and 0 < < 1 is a constant) is NP-hard.

Proof. Follows from Theorem 5. Note that by applying several observations, we can reduce the parameters 372 from Lemma 3 and 98 from the proof of Theorem 5. But this would have no impact on our main results, Theorems 4–6, in the sense of Garey and Johnson [5].

Acknowledgements

The author would like to thank the unknown referees for helpful comments and Bojan Mohar and Mark Ellingham for the discussion on rotational schemes.

References

[1] K. Appel, W. Haken, Every Planar Map is Four Colorable, in: Contemp. Math., vol. 98, Amer. Math. Soc., Providence, 1989. [2] D. Archdeacon, Representativity, in: J.L. Gross, J. Yellen (Eds.), Handbook of Graph Theory, RCR Press, Boca Raton, 2004, pp. 722–736. [3] R. Diestel, Graph Theory, 3rd ed., Springer-Verlag, Heidelberg, 2005. [4] J. Edmonds, A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc. 7 (1960) 646. [5] M.R. Garey, D.S. Johnson, Computers and Intractability, W.H. Freeman, San Francisco, 1979. [6] J.L. Gross, T.W. Tuker, Topological Graph Theory, Wiley, New York, 1987. [7] B. Grünbaum, Conjecture 6, in: W.T. Tutte (Ed.), Recent Progress in Combinatorics, Proceedings of the Third Waterloo Conference on Combinatorics, May 1968, Academic Press, New York, 1969, p. 343. [8] I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718–720. [9] M. Kochol, 3-regular non 3-edge-colorable graphs with polyhedral embeddings in orientable surfaces, in: I.G. Tollis, M. Patrignani (Eds.), Graph Drawing 2008, in: LNCS, vol. 5417, Springer, Berlin, 2009, pp. 319–323. [10] M. Kochol, Polyhedral embeddings of snarks in orientable surfaces, Proc. Amer. Math. Soc. 137 (2009) 1613–1619. [11] M. Kochol, Superposition and constructions of graphs without nowhere-zero k-flows, European J. Combin. 23 (2002) 281–306. [12] M. Kochol, Three measures of edge-uncolorability, manuscript, 2009. [13] M. Kochol, Superposition, Snarks and Flows, EDIS — Žilina Univ. Press, Žilina, Slovakia, 2009. 1860 M. Kochol / Discrete Applied Mathematics 158 (2010) 1856–1860

[14] M. Kochol, N. Krivoňáková, S. Smejová, Edge coloring of multigraphs, Discrete Math. 300 (2005) 229–234. [15] M. Kochol, N. Krivoňáková, S. Smejová, K. Šranková, Complexity of approximation of 3-edge-coloring of graphs, Inform. Process. Lett. 108 (2008) 238–241. [16] B. Mohar, C. Thomassen, Graphs on Surfaces, J. Hopkins Univ. Press, Baltimore, 2001. [17] P.G. Tait, Remarks on the colouring of maps, Proc. Roy. Soc. Edinburgh 10 (1880) 729. [18] A. Vince, Maps, in: J.L. Gross, J. Yellen (Eds.), Handbook of Graph Theory, RCR Press, Boca Raton, 2004, pp. 696–721. [19] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskretn. Anal. 3 (1964) 25–30 (in Russian).