- Email: [email protected]
Subdivision of the hierarchy of H-colorable graph classes by circulant graphs
Electronic Notes in Discrete Mathematics 17 (2004) 269–274 www.elsevier.com/locate/endm
Subdivision of the hierarchy of H-colorable graph classes by circulant graphs Akihiro UEJIMA1 Hiro ITO2 School of Informatics, Kyoto University Yoshida-honmachi, Sakyo-ward, Kyoto 606-8501 Japan
Abstract For any integer p ≥ 2, p-colorable graphs are C2p+1 -colorable and C2p+1 -colorable graphs are p + 1-colorable, where C2p+1 is the complement graph of a cycle of order 2p + 1. The converse statements are however incorrect. This paper presents that the above inclusion can be subdivided by a subset of circulant graphs H(n, k) (n, k ∈ N , k ≤ n/2). The subdivided hierarchy of inclusion contains the well-known inclusion of C2p+1 -colorable graphs. Moreover, we prove some NP-complete problems for planar H(n, k)-colorings, including the C5 -coloring. Keywords: H-coloring; Color-family; Circulant graphs; time complexity
1
Introduction
For a given graph G with vertex set V (G) and edge set E(G), a p-coloring of G is a mapping of V (G) to {1, 2, . . . , p} such that no two vertices on the same edge receive the same color. Given graphs G and H, a homomorphism f of G to H is an edge preserving mapping of V (G) to V (H), i.e., a mapping f of V (G) to V (H) such that f (x) and f (y) are adjacent vertices of H if x and y are adjacent vertices of G. Such a homomorphism is also called an H-coloring of G. For any fixed graph H, H-coloring problem is deciding whether there is an H-coloring of a given input graph G. We say that G is H-colorable if This research is supported by The 21st Century COE Program and the Scientific Grantin-Aid from Ministry of Education, Science, Sports and Culture of Japan. 1 Email: [email protected] 2 Email: [email protected] 1571-0653/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.endm.2004.03.050
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A. Uejima, H. Ito / Electronic Notes in Discrete Mathematics 17 (2004) 269–274
⊂
H(8, 4)
···
⊂
H(9, 4)
⊂
H(5, 2)
⊂
H(6, 3)
⊆ H(8, 3) ⊂
H(7, 3)
⊂
⊂ ···
H(3, 1)
⊂
H(4, 2)
C7
K3
C5
C7
C9
K2
H(2, 1)
H(10, 4)
⊂ H(11, 4) ⊂
···
H(6, 2)
H(9, 3)
H(12, 4)
H(7, 2)
⊂
⊂ ···
⊂ H(10, 3) ⊆ H(11, 3) ⊂ · · ·
⊂ H(13, 4) ⊂
H(14, 4)
···
⊂ H(15, 4) ⊂ · · ·
···
Table 1 Inclusion on H(n, k) and relation with Kp , Cq , and Cr
an H-coloring exists. A complete graph of order n is denoted by Kn . Then, the problem of deciding whether G is Kp -colorable is the problem of deciding whether G is p-colorable. Thus, H-coloring problem is a natural generalization of the traditional graph colorings, and it has been studied in various contexts (2; 5; 6; 7; 8; 9). The interconnections between homomorphisms and the theory of grammar forms were investigated in (5), and H-coloring problems include T -colorings and problems related to channel assignment problems (6). Equivalently, H-coloring problem may be considered as a decision problem related to a class L(H) := {G | G is H-colorable}. Such classes are called color-families and their structure has been an important theme, e.g., Theorems 1.4 and 1.5, which will be shown later. Some terms are defined for explaining the statements we show, as follows. We say that two graphs H and H are homomorphically equivalent if H-colorable and H -colorable are equivalent for any graph. A graph H is a core if it does not admit a homomorphism to a proper subgraph. For example, all complete graphs, odd cycles, and complements of odd cycles are cores. It was proven in (9) that every graph H contains a unique (up to isomorphism) subgraph H which is a core and admits a homomorphism of H to H ; we call H the core of H. For positive integers n and 1 ≤ a1 , a2 , . . . , ak ≤ n/2, the circulant graph (n; a1 , a2 , . . . , ak ) has the vertex set {0, 1, . . . , n − 1}, and two vertices x and y are adjacent if and only if y − x (mod n) or x − y (mod n) ∈ {a1 , a2 , . . . , ak }. We mainly prove two theorems for the subclass of circulant graphs H(n, k) := (n; k, k + 1, . . . , n/2) for 1 ≤ k ≤ n/2 as follows. Corollary 1.2 is obtained from Theorem 1.1. (X ⊂ Y denotes that X is a proper subset of Y , Cp is a cycle of order p, and H is the complement graph of a graph H.) Theorem 1.1 (see Tab. 1) For any integers p ≥ 2, n, k ≥ 1, ◦ L(H(n, k)) ⊆ L(H(n + 1, k)), ◦ L(H(pk − 1, k)) ⊂ L(H(pk, k)) ⊂ L(H(pk + 1, k)) and K p is the core of H(pk, k), and ◦ L(H((2p + 1)k − 1, 2k)) ⊂ L(H((2p + 1)k, 2k)) ⊂ L(H((2p + 1)k + 1, 2k)) and C2p+1 is the core of H((2p + 1)k, 2k).
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Corollary 1.2 For any integers p ≥ 2, L(H(4p, 4)) ⊂ L(H(4p + 1, 4)) ⊂ L(H(4p + 2, 4)) ⊂ L(H(4p + 3, 4)) ⊂ L(H(4(p + 1), 4)). Moreover, K p and C2p+1 are the cores of H(4p, 4) and H(4p + 2, 4), respectively. Theorem 1.3 Planar H(8, 4)-coloring problem is in P. For any 9 ≤ n ≤ 15, planar H(n, 4)-coloring problem is NP-complete. Otherwise, it is in O(1) to solve. Planar C2p+1 -coloring problem is NP-complete for any integer p ≥ 2. Color-families have the already-known basic inclusions as follows (see Tab. 1). The inclusion in Theorem 1.1 is a subdivided hierarchy of these inclusions. Theorem 1.4 (5) For any integers p ≥ 1, q ≥ 2, L(Kp ) ⊂ L(Kp+1 ). Moreover, L(K2 ) ⊂ L(C2q+1 ) ⊂ L(K3 ) = L(C3 ) and L(C2q+1 ) ⊂ L(C2q−1 ). Theorem 1.5 (7) For any integer p ≥ 2, L(Kp ) ⊂ L(C2p+1 ) ⊂ L(Kp+1 ). For general graphs, the complexity of H-coloring problem is well-known, the problem is in P if H is bipartite, otherwise it is NP-complete (2). However, the proofs in (2) cannot be directly extended for planar graphs and it is still open. Needless to say, the problem is in P if H is bipartite. The situation is well-understood for complete graphs: For any fixed p ≥ 4 or p1, the Kp -coloring problem is easy to solve (3), and K2 -coloring problem is in P. K3 -coloring problem is NP-complete (1). In this paper, we are also interested in studying the time complexity of Hcoloring of planar graphs and analyze the threshold graph H such that planar H-coloring problem changes from P to NP-complete preserving a inclusion of color-families. For this purpose, subdividing the inclusion of L(H), e.g., the inclusions in Theorems 1.4 and 1.5 into the one in Theorem 1.1, is not only elegant but also useful for analyzing such a threshold graph H. For the inclusion in Theorem 1.5, the most part of the complexity of planar C2p+1 -coloring problems are solved. For any p ≥ 4, it is in P since L(K4 ) ⊂ L(C2p+1 ). For p = 3, we presented that C7 -coloring problem is NP-complete (8). The case p = 2 is only open (note that it is shown in Theorem 1.3.). We make the time complexity of planar H(n, 4)-coloring problems clear for the new inclusion in Corollary 1.2 and show that L(K2 ) ⊂ L(C2p+1 ), |L(C2p+1 ) − L(K2 )| is boundlessly small, and planar C2p+1 -coloring problem is NP-complete for positive infinity p in Theorem 1.3. We introduce some terms and notations as follows. Consider a graph G and a vertex subset V ⊆ V (G). A graph (V , {(x, y)|x, y ∈ V , (x, y) ∈ E(G)}) is called an induced subgraph of G by V . An induced subgraph of G by V (G)−V is denoted by G − V . If (x, y) ∈ E(G) (resp., (x, y) ∈ / E(G)) for any vertex pair x, y ∈ V , V is called a clique (resp., an independent set). For a graph G, the maximum order of its cliques is denoted by ω(G). An elementary homomorphism in a graph H consists of contracting two non-adjacent vertices
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x and y in H. A graph H is a morphic image of a graph H if it is obtained from H by finitely many elementary homomorphisms. H is also considered to be a morphic image of itself. Theorem 1.6 (5) A graph G is H-colorable if and only if a morphic image G of G is isomorphic to a subgraph of H.
2
Proof of Theorem 1.1
Theorem 1.1 can be directly obtained by the following lemmas. Lemma 2.1 For any integers n ≥ 2, 1 ≤ k ≤ n/2, L(H(n, k)) ⊆ L(H(n + 1, k)). Sketch of proof. Let x be a vertex of V (H(n + 1), k). H(n, k) is a proper subgraph of H(n + 1, k) − {x}. Thus, H(n, k) is H(n + 1, k)-colorable, so all H(n, k)-colorable graphs are H(n + 1, k)-colorable. 2 Lemma 2.2 For any integers n, r ≥ 2, 1 ≤ k ≤ n/2, H(rn, rk) is not a core. That is, H(n, k) and H(rn, rk) are homomorphically equivalent and H(n, k) is a proper subgraph of H(rn, rk). Sketch of proof. A induced subgraph of H(rn, rk) by {0, r, 2r, . . . , (n − 1)r} is H(n, k). Thus, H(n, k) is H(rn, rk)-colorable. On the other hand, Vi := {ri, ri+1, ri+2, . . . , r(i+1)−1} for i ∈ {0, 1, . . . , n− 1} (see Fig. 1(a)). Then, each Vi is a independent set. There is no adjacent vertex-pair between Vi and Vi±j if j ∈ {1, 2, . . . , k − 1}, otherwise there is at least one adjacent vertex-pair between them. Hence, a morphic image of H(rn, rk) constructed by contracting each Vi is H(n, k). From Theorem 1.6, H(rn, rk) is H(n, k)-colorable. 2 Lemma 2.3 For any integers p ≥ 2, 1 ≤ k ≤ n/2, L(H(pk − 1, k)) ⊂ L(H(pk, k)) ⊂ L(H(pk + 1, k)) and L(H((2p + 1)k − 1, 2k)) ⊂ L(H((2p + 1)k, 2k)) ⊂ L(H((2p + 1)k + 1, 2k)). Sketch of proof. We show a sketch of proof about L(H(pk−1, k)) ⊂ L(H(pk, k)) ⊂ L(H(pk +1, k)) (the remaining proof is omitted due to limitations of space). The case L(H(pk − 1, k)) ⊂ L(H(pk, k)): From Lemma 2.2, a core of H(pk, k) is Kp . ω(H(pk − 1, k)) = (pk − 1)/k = p − 1 < p, hence H(pk, k) is not H(pk − 1, k)-colorable. The case L(H(pk, k)) ⊂ L(H(pk + 1, k)): Assume that H(pk + 1, k) is K p colorable (V (Kp ) := {0, 1, . . . , p − 1}). Then, a induced subgraph of H(pk + 1, k) by {0, k, 2k, . . . , (p − 1)k} is Kp . Thus, we can assume that a homomorphism f of H(pk + 1, k) of H(pk, k) such that f (x) = xk , x ∈ {0, k, . . . , (p −
A. Uejima, H. Ito / Electronic Notes in Discrete Mathematics 17 (2004) 269–274
273
1)/k} without loss of generality. The one by {0, k, 2k, . . . , (p−2)k, (p−1)k+1} is also Kp , so f ((p − 1)k + 1) = p − 1 (see Fig. 1). After pk + 1 steps of same operation, f (0) = p − 1, it contradicts the assumption f (0) = 0. 2 Vn-1 (n-1)r
V0
0
r-1 r
0
12
(n-2)r
1
11
2
0
Vn-k kr
(n-k)r
Vk H(rn, rk)
10
K3
3
9 2
4
1
8
Vi
f (8)=2
H(12, 4) = (12; 4, 5, 6)
ir
(i+1)r
(a)
5 7
6
H(13, 4) = (13; 4, 5, 6) (b)
Fig. 1. Sketch of proofs
3
Proof of Theorem 1.3
For this proof, we can reduce Planar 3SAT (4) to the problems (the proofs are omitted). The reductions are similar to the reduction used to prove that planar 3-coloring problem is NP-complete (1).
4
Concluding remarks
This paper was interested in studying the time complexity of H-coloring of planar graphs. We mainly showed a new inclusion of H(n, k)-colorable graphs and the time complexity of some planar H(n, k)-coloring problems.
References [1] M. R. Garey, D. S. Johnson, & L. Stockmeyer, “Some simplified NPcomplete graph problems,” Theoretical Computer Science, 1, pp. 237-267, 1976. [2] P. Hell, & J. Nesetril, “On the Complexity of H-Coloring,” Journal of Combinatorial Theory, B 48, pp. 92-110, 1990. [3] T. R. Jensen, & B. Toft, Graph Coloring Problems, John Wiley & Sons, New York, 1995.
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[4] D. Lichtenstein, “Planar formulae and their uses,” SIAM J. Comput., 11, no. 2, pp. 329-343, 1982. [5] H. A. Maurer, A. Salomaa, & D. Wood, “Colorings and interpretations : A connection between graphs and grammar forms,” Discrete Applied Mathematics, 3, pp. 119-135, 1981. [6] F. S. Roberts, “T -colorings of graphs: recent results and open problems,” Discrete Math., 93, pp. 229–245, 1991. [7] A. Uejima, and H. Ito, “On H-coloring problems with H expressed by complements of cycles, bipartite graphs, and chordal graphs,” IEICE Transactions, vol. E85-A, no. 5, pp. 1026–1030, 2002. [8] A. Uejima, H. Ito, and T. Tsukiji, “C7 -coloring problem,” IEICE Transactions, 2004 (to appear). [9] E. Welzl, “Color families are dense,” Theoret. Comput. Sci., 17, pp. 29-41, 1970.
Subdivision of the hierarchy of H-colorable graph classes by circulant graphs Akihiro UEJIMA1 Hiro ITO2 School of Informatics, Kyoto University Yoshida-honmachi, Sakyo-ward, Kyoto 606-8501 Japan
Abstract For any integer p ≥ 2, p-colorable graphs are C2p+1 -colorable and C2p+1 -colorable graphs are p + 1-colorable, where C2p+1 is the complement graph of a cycle of order 2p + 1. The converse statements are however incorrect. This paper presents that the above inclusion can be subdivided by a subset of circulant graphs H(n, k) (n, k ∈ N , k ≤ n/2). The subdivided hierarchy of inclusion contains the well-known inclusion of C2p+1 -colorable graphs. Moreover, we prove some NP-complete problems for planar H(n, k)-colorings, including the C5 -coloring. Keywords: H-coloring; Color-family; Circulant graphs; time complexity
1
Introduction
For a given graph G with vertex set V (G) and edge set E(G), a p-coloring of G is a mapping of V (G) to {1, 2, . . . , p} such that no two vertices on the same edge receive the same color. Given graphs G and H, a homomorphism f of G to H is an edge preserving mapping of V (G) to V (H), i.e., a mapping f of V (G) to V (H) such that f (x) and f (y) are adjacent vertices of H if x and y are adjacent vertices of G. Such a homomorphism is also called an H-coloring of G. For any fixed graph H, H-coloring problem is deciding whether there is an H-coloring of a given input graph G. We say that G is H-colorable if This research is supported by The 21st Century COE Program and the Scientific Grantin-Aid from Ministry of Education, Science, Sports and Culture of Japan. 1 Email: [email protected] 2 Email: [email protected] 1571-0653/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.endm.2004.03.050
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A. Uejima, H. Ito / Electronic Notes in Discrete Mathematics 17 (2004) 269–274
⊂
H(8, 4)
···
⊂
H(9, 4)
⊂
H(5, 2)
⊂
H(6, 3)
⊆ H(8, 3) ⊂
H(7, 3)
⊂
⊂ ···
H(3, 1)
⊂
H(4, 2)
C7
K3
C5
C7
C9
K2
H(2, 1)
H(10, 4)
⊂ H(11, 4) ⊂
···
H(6, 2)
H(9, 3)
H(12, 4)
H(7, 2)
⊂
⊂ ···
⊂ H(10, 3) ⊆ H(11, 3) ⊂ · · ·
⊂ H(13, 4) ⊂
H(14, 4)
···
⊂ H(15, 4) ⊂ · · ·
···
Table 1 Inclusion on H(n, k) and relation with Kp , Cq , and Cr
an H-coloring exists. A complete graph of order n is denoted by Kn . Then, the problem of deciding whether G is Kp -colorable is the problem of deciding whether G is p-colorable. Thus, H-coloring problem is a natural generalization of the traditional graph colorings, and it has been studied in various contexts (2; 5; 6; 7; 8; 9). The interconnections between homomorphisms and the theory of grammar forms were investigated in (5), and H-coloring problems include T -colorings and problems related to channel assignment problems (6). Equivalently, H-coloring problem may be considered as a decision problem related to a class L(H) := {G | G is H-colorable}. Such classes are called color-families and their structure has been an important theme, e.g., Theorems 1.4 and 1.5, which will be shown later. Some terms are defined for explaining the statements we show, as follows. We say that two graphs H and H are homomorphically equivalent if H-colorable and H -colorable are equivalent for any graph. A graph H is a core if it does not admit a homomorphism to a proper subgraph. For example, all complete graphs, odd cycles, and complements of odd cycles are cores. It was proven in (9) that every graph H contains a unique (up to isomorphism) subgraph H which is a core and admits a homomorphism of H to H ; we call H the core of H. For positive integers n and 1 ≤ a1 , a2 , . . . , ak ≤ n/2, the circulant graph (n; a1 , a2 , . . . , ak ) has the vertex set {0, 1, . . . , n − 1}, and two vertices x and y are adjacent if and only if y − x (mod n) or x − y (mod n) ∈ {a1 , a2 , . . . , ak }. We mainly prove two theorems for the subclass of circulant graphs H(n, k) := (n; k, k + 1, . . . , n/2) for 1 ≤ k ≤ n/2 as follows. Corollary 1.2 is obtained from Theorem 1.1. (X ⊂ Y denotes that X is a proper subset of Y , Cp is a cycle of order p, and H is the complement graph of a graph H.) Theorem 1.1 (see Tab. 1) For any integers p ≥ 2, n, k ≥ 1, ◦ L(H(n, k)) ⊆ L(H(n + 1, k)), ◦ L(H(pk − 1, k)) ⊂ L(H(pk, k)) ⊂ L(H(pk + 1, k)) and K p is the core of H(pk, k), and ◦ L(H((2p + 1)k − 1, 2k)) ⊂ L(H((2p + 1)k, 2k)) ⊂ L(H((2p + 1)k + 1, 2k)) and C2p+1 is the core of H((2p + 1)k, 2k).
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Corollary 1.2 For any integers p ≥ 2, L(H(4p, 4)) ⊂ L(H(4p + 1, 4)) ⊂ L(H(4p + 2, 4)) ⊂ L(H(4p + 3, 4)) ⊂ L(H(4(p + 1), 4)). Moreover, K p and C2p+1 are the cores of H(4p, 4) and H(4p + 2, 4), respectively. Theorem 1.3 Planar H(8, 4)-coloring problem is in P. For any 9 ≤ n ≤ 15, planar H(n, 4)-coloring problem is NP-complete. Otherwise, it is in O(1) to solve. Planar C2p+1 -coloring problem is NP-complete for any integer p ≥ 2. Color-families have the already-known basic inclusions as follows (see Tab. 1). The inclusion in Theorem 1.1 is a subdivided hierarchy of these inclusions. Theorem 1.4 (5) For any integers p ≥ 1, q ≥ 2, L(Kp ) ⊂ L(Kp+1 ). Moreover, L(K2 ) ⊂ L(C2q+1 ) ⊂ L(K3 ) = L(C3 ) and L(C2q+1 ) ⊂ L(C2q−1 ). Theorem 1.5 (7) For any integer p ≥ 2, L(Kp ) ⊂ L(C2p+1 ) ⊂ L(Kp+1 ). For general graphs, the complexity of H-coloring problem is well-known, the problem is in P if H is bipartite, otherwise it is NP-complete (2). However, the proofs in (2) cannot be directly extended for planar graphs and it is still open. Needless to say, the problem is in P if H is bipartite. The situation is well-understood for complete graphs: For any fixed p ≥ 4 or p1, the Kp -coloring problem is easy to solve (3), and K2 -coloring problem is in P. K3 -coloring problem is NP-complete (1). In this paper, we are also interested in studying the time complexity of Hcoloring of planar graphs and analyze the threshold graph H such that planar H-coloring problem changes from P to NP-complete preserving a inclusion of color-families. For this purpose, subdividing the inclusion of L(H), e.g., the inclusions in Theorems 1.4 and 1.5 into the one in Theorem 1.1, is not only elegant but also useful for analyzing such a threshold graph H. For the inclusion in Theorem 1.5, the most part of the complexity of planar C2p+1 -coloring problems are solved. For any p ≥ 4, it is in P since L(K4 ) ⊂ L(C2p+1 ). For p = 3, we presented that C7 -coloring problem is NP-complete (8). The case p = 2 is only open (note that it is shown in Theorem 1.3.). We make the time complexity of planar H(n, 4)-coloring problems clear for the new inclusion in Corollary 1.2 and show that L(K2 ) ⊂ L(C2p+1 ), |L(C2p+1 ) − L(K2 )| is boundlessly small, and planar C2p+1 -coloring problem is NP-complete for positive infinity p in Theorem 1.3. We introduce some terms and notations as follows. Consider a graph G and a vertex subset V ⊆ V (G). A graph (V , {(x, y)|x, y ∈ V , (x, y) ∈ E(G)}) is called an induced subgraph of G by V . An induced subgraph of G by V (G)−V is denoted by G − V . If (x, y) ∈ E(G) (resp., (x, y) ∈ / E(G)) for any vertex pair x, y ∈ V , V is called a clique (resp., an independent set). For a graph G, the maximum order of its cliques is denoted by ω(G). An elementary homomorphism in a graph H consists of contracting two non-adjacent vertices
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A. Uejima, H. Ito / Electronic Notes in Discrete Mathematics 17 (2004) 269–274
x and y in H. A graph H is a morphic image of a graph H if it is obtained from H by finitely many elementary homomorphisms. H is also considered to be a morphic image of itself. Theorem 1.6 (5) A graph G is H-colorable if and only if a morphic image G of G is isomorphic to a subgraph of H.
2
Proof of Theorem 1.1
Theorem 1.1 can be directly obtained by the following lemmas. Lemma 2.1 For any integers n ≥ 2, 1 ≤ k ≤ n/2, L(H(n, k)) ⊆ L(H(n + 1, k)). Sketch of proof. Let x be a vertex of V (H(n + 1), k). H(n, k) is a proper subgraph of H(n + 1, k) − {x}. Thus, H(n, k) is H(n + 1, k)-colorable, so all H(n, k)-colorable graphs are H(n + 1, k)-colorable. 2 Lemma 2.2 For any integers n, r ≥ 2, 1 ≤ k ≤ n/2, H(rn, rk) is not a core. That is, H(n, k) and H(rn, rk) are homomorphically equivalent and H(n, k) is a proper subgraph of H(rn, rk). Sketch of proof. A induced subgraph of H(rn, rk) by {0, r, 2r, . . . , (n − 1)r} is H(n, k). Thus, H(n, k) is H(rn, rk)-colorable. On the other hand, Vi := {ri, ri+1, ri+2, . . . , r(i+1)−1} for i ∈ {0, 1, . . . , n− 1} (see Fig. 1(a)). Then, each Vi is a independent set. There is no adjacent vertex-pair between Vi and Vi±j if j ∈ {1, 2, . . . , k − 1}, otherwise there is at least one adjacent vertex-pair between them. Hence, a morphic image of H(rn, rk) constructed by contracting each Vi is H(n, k). From Theorem 1.6, H(rn, rk) is H(n, k)-colorable. 2 Lemma 2.3 For any integers p ≥ 2, 1 ≤ k ≤ n/2, L(H(pk − 1, k)) ⊂ L(H(pk, k)) ⊂ L(H(pk + 1, k)) and L(H((2p + 1)k − 1, 2k)) ⊂ L(H((2p + 1)k, 2k)) ⊂ L(H((2p + 1)k + 1, 2k)). Sketch of proof. We show a sketch of proof about L(H(pk−1, k)) ⊂ L(H(pk, k)) ⊂ L(H(pk +1, k)) (the remaining proof is omitted due to limitations of space). The case L(H(pk − 1, k)) ⊂ L(H(pk, k)): From Lemma 2.2, a core of H(pk, k) is Kp . ω(H(pk − 1, k)) = (pk − 1)/k = p − 1 < p, hence H(pk, k) is not H(pk − 1, k)-colorable. The case L(H(pk, k)) ⊂ L(H(pk + 1, k)): Assume that H(pk + 1, k) is K p colorable (V (Kp ) := {0, 1, . . . , p − 1}). Then, a induced subgraph of H(pk + 1, k) by {0, k, 2k, . . . , (p − 1)k} is Kp . Thus, we can assume that a homomorphism f of H(pk + 1, k) of H(pk, k) such that f (x) = xk , x ∈ {0, k, . . . , (p −
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273
1)/k} without loss of generality. The one by {0, k, 2k, . . . , (p−2)k, (p−1)k+1} is also Kp , so f ((p − 1)k + 1) = p − 1 (see Fig. 1). After pk + 1 steps of same operation, f (0) = p − 1, it contradicts the assumption f (0) = 0. 2 Vn-1 (n-1)r
V0
0
r-1 r
0
12
(n-2)r
1
11
2
0
Vn-k kr
(n-k)r
Vk H(rn, rk)
10
K3
3
9 2
4
1
8
Vi
f (8)=2
H(12, 4) = (12; 4, 5, 6)
ir
(i+1)r
(a)
5 7
6
H(13, 4) = (13; 4, 5, 6) (b)
Fig. 1. Sketch of proofs
3
Proof of Theorem 1.3
For this proof, we can reduce Planar 3SAT (4) to the problems (the proofs are omitted). The reductions are similar to the reduction used to prove that planar 3-coloring problem is NP-complete (1).
4
Concluding remarks
This paper was interested in studying the time complexity of H-coloring of planar graphs. We mainly showed a new inclusion of H(n, k)-colorable graphs and the time complexity of some planar H(n, k)-coloring problems.
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