- Email: [email protected]
Edge colorings of planar graphs without 5-cycles with two chords
Theoretical Computer Science 518 (2014) 124–127
Contents lists available at SciVerse ScienceDirect
Theoretical Computer Science www.elsevier.com/locate/tcs
Note
Edge colorings of planar graphs without 5-cycles with two chords ✩ Jian-Liang Wu a,∗ , Ling Xue b a b
School of Mathematics, Shandong University, Jinan, 250100, China Department of Information Engineering, Taishan Polytechnic, Tai’an, 271000, China
a r t i c l e
i n f o
Article history: Received 4 January 2013 Received in revised form 16 July 2013 Accepted 18 July 2013 Communicated by D.-Z. Du
a b s t r a c t A graph G is of class 1 if its edges can be colored with k colors in such a way that adjacent edges receive different colors, where k is the maximum degree of G. It is proved here that every planar graph is of class 1 if its maximum degree is at least 6 and any 5-cycle contains at most one chord. © 2013 Elsevier B.V. All rights reserved.
Keywords: Edge coloring Planar graph Cycle Class 1
1. Introduction All graphs considered here are finite and simple. Let G be a graph with the vertex set V (G ) and edge set E (G ). If v ∈ V (G ), then its neighbor set N G ( v ) (or simply N ( v )) is the set of the vertices in G adjacent to v and the degree d( v ) of v is | N G ( v )|. We denote the maximum degree of G by (G ). For V ⊆ V (G ), denote N ( V ) = u ∈ V N (u ). A k-, k+ -vertex is a vertex of degree k, at least k. A k (or k+ )-vertex adjacent to a vertex x is called a k (or k+ )-neighbor of x. Let dk (x), dk+ (x) denote the number of k-neighbors, k+ -neighbors of x. A k-cycle is a cycle of length k. Two cycles sharing a common edge are said to be adjacent. Given a cycle C of length k in G, an edge xy ∈ E (G )\ E (C ) is called a chord of C if x, y ∈ V (C ). Such a cycle C is also called a chordal-k-cycle. A graph is k-edge-colorable, if its edges can be colored with k colors in such a way that adjacent edges receive different colors. The edge chromatic number of a graph G, denoted by χ (G ), is the smallest integer k such that G is k-edge-colorable. In 1964, Vizing showed that if G is a graph with maximum degree , then (G ) χ (G ) (G ) + 1. A graph G is said to be of class 1 if χ (G ) = , and of class 2 if χ (G ) = + 1. A graph G is critical if it is connected and of class 2, and χ (G − e) < χ (G ) for any edge e of G. A critical graph with maximum degree is called a -critical graph. It is clear that every critical graph is 2-connected. For planar graphs, more is known. As noted by Vizing [2], if C 4 , K 4 , the octahedron, and the icosahedron have one edge subdivided each, class 2 planar graphs are produced for ∈ {2, 3, 4, 5}. He proved that every planar graph with 8 is of class 1 (there are more general results, see [3] and [5]) and then conjectured that every planar graph with maximum degree 6 or 7 is of class 1. The case = 7 for the conjecture has been verified by Zhang [9] and, independently, by Sanders and Zhao [6]. The case = 6 remains open, but some partial results are obtained. Theorem 16.3 [2] stated that a planar
✩
*
This work was partially supported by National Natural Science Foundation of China (No. 11271006). Corresponding author. E-mail address: [email protected] (J.-L. Wu).
0304-3975/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tcs.2013.07.027
J.-L. Wu, L. Xue / Theoretical Computer Science 518 (2014) 124–127
125
graph with the maximum degree and the girth g is of class 1 if 3 and g 8, or 4 and g 5, or 5 and g 4. Lam, Liu, Shiu and Wu [4] proved that a planar graph G is of class 1 if 6 and any vertex is incident with at most one 3-cycle. Zhou [10] obtained that every planar graph with 6 and without 4- or 5-cycles is of class 1. Bu and Wang [1] proved that every planar graph with 6 and without chordal 5-cycles and chordal 6-cycles is of class 1. Wang and Chen [7] proved that every planar graph is of class 1 if 6 and it does not contain a 5-cycle with a chord. In the paper, we shall improve the above result by proving that every planar graph with = 6 and without 5-cycles with two chords is of class 1. Recently, Wang and Xu [8] proved that every plane graph G with maximum degree 6 is edge 6-colorable if no vertex in G is incident with four faces of size 3. 2. The main result and its proof To prove our result, we will introduce some known lemmas. Lemma 1. (See [6,9].) If G is a planar graph with (G ) 7, then G is of class 1. Lemma 2. (Vizing’s Adjacency Lemma [2]). Let G be a -critical graph, and let u and v be adjacent vertices of G with d( v ) = k. (a) If k < , then u is adjacent to at least − k + 1 vertices of degree ; (b) If k = , then u is adjacent to at least two vertices of degree . From Vizing’s Adjacency Lemma, it is easy to get the following corollary. Corollary 3. Let G be a -critical graph. Then (a) every vertex is adjacent to at most one 2-vertex and at least two -vertices; (b) the sum of the degree of any two adjacent vertices is at least + 2; (c) if uv ∈ E (G ) and d(u ) + d( v ) = + 2, then every vertex of N ({u , v }) \ {u , v } is a -vertex. Lemma 4. (See [9].) Let G be a -critical graph, uv ∈ E (G ) and d(u ) + d( v ) = + 2. Then (a) every vertex of N ( N ({u , v })) \ {u , v } is of degree at least − 1; (b) if d(u ), d( v ) < , then every vertex of N ( N ({u , v })) \ {u , v } is a -vertex. Lemma 5. (See [6].) No -critical graph has distinct vertices x, y, z such that x is adjacent to y and z, d( z) < 2 − d(x) − d( y ) + 2, and xz is in at least d(x) + d( y ) − − 2 triangles not containing y. To be convenient, we give some definitions and notations on planar graphs. Let G be a plane graph and F (G ) the face set of G. A face of G is said to be incident with all edges and vertices in its boundary. Two faces sharing an edge e are said to be adjacent at e. The degree of a face f of G, denoted by d G ( f ), is the number of edges incident with f where each cut edge is counted twice. A k-, k+ -face is a face of degree k, at least k. A k-face of G is called an (i 1 , i 2 , . . . , ik )-face if the vertices in its boundary are of degrees i 1 , i 2 , . . . , ik respectively. A 3-face is denoted by [x, y , z] if it is incident with distinct vertices x, y , z and d(x) d( y ) d( z). For a vertex v ∈ V (G ), we denote by f k ( v ) the number of k-faces incident with v. Theorem 6. Let G be a planar graph with 6. If any 5-cycle contains at most one chord, then G is of class 1. Proof. Suppose that G is a counterexample to our theorem with the minimum number of edges and suppose that G is embedded in the plane. Then G is a 6-critical graph by Lemma 1, and it is 2-connected and Lemma 2. By Euler’s formula | V (G )| − | E (G )| + | F (G )| = 2, we have
d(x) − 4 +
x∈ V (G )
d(x) − 4 = −8 < 0.
x∈ F (G )
We define ch to be the initial charge. Let ch(x) = d(x) − 4 for each x ∈ V ∪ F . So x∈ V ∪ F ch(x) < 0. In the following, we will reassign a new charge denoted by ch (x) to each x ∈ V ∪ F according to the discharging rules. Since our rules only move charges around, and do not affect the sum. If we can show that ch (x) 0 for each x ∈ V ∪ F , then we get an obvious contradiction 0 x∈ V ∪ F ch (x) = x∈ V ∪ F ch(x) < 0, which completes our proof. A 4-face f = [ w , v , x, y ] is called special if d(x) = 2 and v , x, y form a 3-face. The discharging rules are defined as follows. R1 Let v be a 2-vertex. If v is incident with at least one 5+ -face, then v receives 1 from any of its incident 5+ -face, from each adjacent vertex; Otherwise, if v is incident with a special 4-face f , then v receives adjacent vertex; Otherwise v receives 1 from each adjacent vertex.
1 3
from f ,
5 6
1 2
from each
126
J.-L. Wu, L. Xue / Theoretical Computer Science 518 (2014) 124–127
R2 Every 3-vertex receives
1 3
from each adjacent vertex.
R3 Let f be a 3-face [x, y , z] such that d(x) d( y ) d( z). If 2 d(x) 4 and d( y ) 5, then f receives 1 3
1 2
from y,
1 2
from
z; If d(x) = d( y ) = 4 and d( z) = 6, then f receives 1 from z; If d(x) 5, then f receives from x, y , z, respectively. R4 Let v be a 5-vertex. R4.1 If v is adjacent to a 6-vertex x and incident with a (3, 5, 6)-face [u , v , w ] such that ux ∈ / E (G ) and w = x, then v receives 13 from x; R4.2 Suppose that v is adjacent to a 4-vertex and incident with two adjacent 3-faces uv w and v wx. Then d6 ( v ) 3. If max{d(u ), d(x)} 5, then v receives 13 from w; Otherwise, choose a vertex y ∈ {u , x} with d( y ) = 6. If d( w ) = 4, then v receives 13 from y; Otherwise v receives 16 from y. R5 Let v be a 6-vertex. R5.1 If v is incident with a special 4-face f = [ w , v , x, y ] such that d( y ) = 2, then v sends
1 3
to f ;
R5.2 If v is incident with two 3-faces [u , v , x] and [ v , x, y ] such that d(x) = d( y ) = 4, then v receives
1 6
from u.
Now, let’s begin to check ch (x) 0 for all x ∈ V ∪ F . Let f ∈ F (G ). Then d( f ) 3. If d( f ) 5, then the number of 2-vertices incident with f is at most d( f ) − 4, and it follows that ch ( f ) ch( f ) − (d( f ) − 4) = 0 by R1. Suppose d( f ) = 4. If f is special, then ch ( f ) = 0 + 13 − 13 = 0 by R1 and R5; Otherwise, ch ( f ) = ch( f ) = 0. Suppose d( f ) = 3. Since = 6, f must be the (2+ , 6, 6)-face, (3, 5+ , 6)-face, (4, 4, 6)-face or (4+ , 5, 5)-face by Lemma 2. Hence ch ( f ) = ch( f ) + max{2 × 1 , 1, 3 × 13 } = 0 by R3. 2 Let w ∈ V (G ). Then d( w ) 2. If d( w ) = 2, then w is adjacent to two 6-vertices by Corollary 3, so ch ( w ) = ch( w ) + max{1 + 2 × 12 , 12 + 2 × 56 , 2 × 1} = 0 by R1. If d( w ) = 3, then w is adjacent to three 5+ -vertices by Corollary 3, and it
follows that ch ( w ) = −1 + 3 × 13 = 0 by R2. If d( w ) = 4, then ch ( w ) = ch( w ) = 0. Suppose that d( w ) = 5. We have ch( w ) = 1, f 3 ( w ) 3, min{d(u ) | u ∈ N ( w )} 3, d3 ( w ) 1 and d6 ( w ) 2. Let w 0 , w 1 , . . . , w 4 be neighbors of w and f 0 , f 1 , . . . , f 4 be faces incident with w such that f i is incident with w i and w i +1 , for all i ∈ {0, 1, . . . , 4}, where w 5 = w 0 . If all neighbors of w are 5+ -vertices, then ch ( w ) 1 − 3 × 13 = 0 by R3. Suppose that min0i 4 d( w i ) = 4. If f 3 ( w ) 2, then ch ( w ) 1 − 2 ×
1 2
= 0 by R3; Otherwise, without loss of generality, 1 + 13 ) 2
assume that f 4 , f 0 , f 2 are 3-faces. If d( w 0 ) = 4, then d( w i ) 5 (1 i 4) by Lemma 5. So w sends at most (2 × to its adjacent 3-faces. At the same time, w receives
ch ( w ) 1 +
1 3
− (2 ×
1 2
1 3
from one of its adjacent 6-vertices by R4.2, and it follows that
= 0. The case that d( w 1 ) = 4 or d( w 2 ) = 4 can be similarly checked as above. Suppose that d3 ( w ) = 1, without loss of generality, assume that d( w 1 ) = 3. Then w sends 13 to w 1 by R2 and d6 ( w ) = 4 by Lemma 2. If f 3 ( w ) 1 or w w 1 is not incident with a 3-face, then ch ( w ) 1 − 13 − max{ 12 , 2 × 13 } = 0 by R2 and R3; Otherwise, f 3 ( w ) 2 and w w 1 is incident with a 3-face. Without loss of generality, assume that d( f 1 ) = 3. By R4.1, w 3 , w 4 send 13 to w, respectively. So ch ( w ) 1 + 2 × 13 − 13 − 2 × 12 − 13 = 0 by R2 and R3 (note that f 0 may be also a 3-face). In the following we check the case that d( w ) = 6. Thus we have ch( w ) = 2, f 3 ( w ) 4, d2 ( w ) 1 and d6 ( w ) 2 by Lemma 2. Note that if w is incident with a special 4-face f = [ w , v , x, y ] such that d(x) = 2, then two faces adjacent to f and incident with w are 4+ -faces since G contains no 5-cycles with two chords. We denote by f s4 ( w ) the number of special 4-faces incident with w and f 34 = f 3 ( w ) + f s4 ( w ). So f s4 ( w ) 3, f 34 ( w ) 4 and if f s4 ( w ) 2, then f 3 ( w ) 1. Case 1. w sends
1 3
1 3
+
1 ) 3
to some adjacent 5-vertex v (ref. R4).
Suppose that v is incident with a (3, 5, 6)-face [u , v , x] such that wu ∈ / E (G ) and w = x (see R4.1). Then w may send to v by R4.1. At the same time, w is adjacent to five 6-vertices by Lemma 4, that is, d6 ( w ) = 5. Since f 34 ( w ) 4,
ch ( w ) = 2 − 13 − 4 × 13 > 0. Suppose that d4 ( v ) 1 and v is incident with three 3-faces [ v , v 0 , v 1 ], [ v , v 1 , v 2 ], [ v , v 3 , v 4 ] (see R4.2). Then d6 ( v ) 3. If w = v 1 , then min{d( v 0 ), d( v 2 )} = 4, max{d( v 0 ), d( v 2 )} 5, f 6 ( w ) = 3, and it follows that ch ( w ) 2 − 13 − 2 × 12 − 2 × 13 = 0; Otherwise, w ∈ { v 0 , v 2 } and f 34 ( w ) 3. Since d( v ) = 5, any neighbor of w is the 3+ -vertex and incident with no (3, 5, 6)-faces and (4, 4, 6)-faces. So ch ( w ) 2 − 2 × 16 − 3 × 12 > 0. Case 2. w sends
1 6
to some adjacent 6-vertex v (ref. R5.2).
Then v is incident with two 3-faces [ w , v , x] and [ v , x, y ] such that d(x) = d( y ) = 4. Then f 34 ( w ) 3 and d6 ( w ) = 5 by Lemma 4. So ch ( w ) = 2 − 16 − 12 − 2 × 13 > 0. Case 3. w sends no charge to its adjacent 5+ -vertices. Let k = min{d(u ) | u ∈ N ( w )}. If k 5, then ch ( w ) 2 − 4 × 13 > 0. Suppose that k = 4. Then d6 ( w ) 3 by Lemma 2. If w is incident with two 3-faces [u , w , x] and [ w , x, y ] such that d(x) = d( y ) = 4, then d6 ( w ) = 4, w y is incident with
J.-L. Wu, L. Xue / Theoretical Computer Science 518 (2014) 124–127
a 4+ -face, w receives
1 6
from u, and it follows that ch ( w ) 2 +
1 6
1 2
−2×
1 3
= 0 since f 34 ( w ) 4; Otherwise, = 0. Suppose that k = 3. Then d6 ( w ) 4 by Lemma 2. If d3 ( w ) = 1 and d5+ ( w ) 5, then ch ( w ) 2 − 13 − 2 × ( 12 + 13 ) = 0;
ch ( w ) 2 − max{1 + 2 ×
1 , 4 × 12 } 3
−1−
127
Otherwise, w is incident with two 4− -vertices u, v, then u and v are incident with at most one 3-face by Lemma 5 since d(u ) + d( v ) + d( w ) 3 + 4 + 6 < 14. So f 34 ( w ) 3, and it follows that ch ( w ) 2 − 13 − 12 − 2 × 13 > 0 by R1 and R3. Suppose that k = 2, that is, w is adjacent to a 2-vertex v. Then d6 ( w ) = 5 by Lemma 2. If v is incident with a special 4-face f = [u , v , w , x], then f 3 ( v ) 3 and w sends 56 to v, and it follows that ch ( w ) 2 − 56 − 12 − 2 × 13 = 0; Otherwise, v is incident with a 5+ -face or two 4-faces. If v is incident with a 5+ -face, then w sends 2−
1 2
− ( 12
+3×
1 ) 3
1 2
to v, and it follows that ch ( w )
= 0. If v is incident with two 4-faces, then f 34 3, and it follows that ch ( w ) 2 − 1 − 3 ×
1 3
= 0. 2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Y.H. Bu, W.F. Wang, Some sufficient conditions for a planar graph of maximum degree six to be class 1, Discrete Math. 306 (13) (2006) 1440–1445. S. Fiorini, R.J. Wilson, Edge-Colorings of Graphs, Research Notes in Mathematics, vol. 16, Pitman, London, 1977. H. Hind, Y. Zhao, Edge colorings of graphs embeddable in a surface of low genus, Discrete Math. 190 (1998) 107–114. P. Lam, J. Liu, W. Shiu, J. Wu, Some sufficient conditions for a planar graph to be of Class 1, Congr. Numer. 136 (1999) 201–205. L.Y. Miao, J.L. Wu, Edge-coloring critical graphs with high degree, Discrete Math. 257 (1) (2002) 169–172. D.P. Sanders, Y. Zhao, Planar graphs of maximum degree seven are class 1, J. Combin. Theory Ser. B 83 (2001) 202–212. W.F. Wang, Y.Z. Chen, A sufficient condition for a planar graph to be class 1, Theoret. Comput. Sci. 385 (1–3) (2007) 71–77. Y.Q. Wang, L.J. Xu, A sufficient condition for a plane graph with maximum degree 6 to be class 1, Discrete Appl. Math. 161 (2013) 307–310. L.M. Zhang, Every planar graph with maximum degree 7 is of class 1, Graphs Combin. 16 (2000) 467–495. G.F. Zhou, A note on graphs of class 1, Discrete Math. 263 (2003) 339–345.
Contents lists available at SciVerse ScienceDirect
Theoretical Computer Science www.elsevier.com/locate/tcs
Note
Edge colorings of planar graphs without 5-cycles with two chords ✩ Jian-Liang Wu a,∗ , Ling Xue b a b
School of Mathematics, Shandong University, Jinan, 250100, China Department of Information Engineering, Taishan Polytechnic, Tai’an, 271000, China
a r t i c l e
i n f o
Article history: Received 4 January 2013 Received in revised form 16 July 2013 Accepted 18 July 2013 Communicated by D.-Z. Du
a b s t r a c t A graph G is of class 1 if its edges can be colored with k colors in such a way that adjacent edges receive different colors, where k is the maximum degree of G. It is proved here that every planar graph is of class 1 if its maximum degree is at least 6 and any 5-cycle contains at most one chord. © 2013 Elsevier B.V. All rights reserved.
Keywords: Edge coloring Planar graph Cycle Class 1
1. Introduction All graphs considered here are finite and simple. Let G be a graph with the vertex set V (G ) and edge set E (G ). If v ∈ V (G ), then its neighbor set N G ( v ) (or simply N ( v )) is the set of the vertices in G adjacent to v and the degree d( v ) of v is | N G ( v )|. We denote the maximum degree of G by (G ). For V ⊆ V (G ), denote N ( V ) = u ∈ V N (u ). A k-, k+ -vertex is a vertex of degree k, at least k. A k (or k+ )-vertex adjacent to a vertex x is called a k (or k+ )-neighbor of x. Let dk (x), dk+ (x) denote the number of k-neighbors, k+ -neighbors of x. A k-cycle is a cycle of length k. Two cycles sharing a common edge are said to be adjacent. Given a cycle C of length k in G, an edge xy ∈ E (G )\ E (C ) is called a chord of C if x, y ∈ V (C ). Such a cycle C is also called a chordal-k-cycle. A graph is k-edge-colorable, if its edges can be colored with k colors in such a way that adjacent edges receive different colors. The edge chromatic number of a graph G, denoted by χ (G ), is the smallest integer k such that G is k-edge-colorable. In 1964, Vizing showed that if G is a graph with maximum degree , then (G ) χ (G ) (G ) + 1. A graph G is said to be of class 1 if χ (G ) = , and of class 2 if χ (G ) = + 1. A graph G is critical if it is connected and of class 2, and χ (G − e) < χ (G ) for any edge e of G. A critical graph with maximum degree is called a -critical graph. It is clear that every critical graph is 2-connected. For planar graphs, more is known. As noted by Vizing [2], if C 4 , K 4 , the octahedron, and the icosahedron have one edge subdivided each, class 2 planar graphs are produced for ∈ {2, 3, 4, 5}. He proved that every planar graph with 8 is of class 1 (there are more general results, see [3] and [5]) and then conjectured that every planar graph with maximum degree 6 or 7 is of class 1. The case = 7 for the conjecture has been verified by Zhang [9] and, independently, by Sanders and Zhao [6]. The case = 6 remains open, but some partial results are obtained. Theorem 16.3 [2] stated that a planar
✩
*
This work was partially supported by National Natural Science Foundation of China (No. 11271006). Corresponding author. E-mail address: [email protected] (J.-L. Wu).
0304-3975/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tcs.2013.07.027
J.-L. Wu, L. Xue / Theoretical Computer Science 518 (2014) 124–127
125
graph with the maximum degree and the girth g is of class 1 if 3 and g 8, or 4 and g 5, or 5 and g 4. Lam, Liu, Shiu and Wu [4] proved that a planar graph G is of class 1 if 6 and any vertex is incident with at most one 3-cycle. Zhou [10] obtained that every planar graph with 6 and without 4- or 5-cycles is of class 1. Bu and Wang [1] proved that every planar graph with 6 and without chordal 5-cycles and chordal 6-cycles is of class 1. Wang and Chen [7] proved that every planar graph is of class 1 if 6 and it does not contain a 5-cycle with a chord. In the paper, we shall improve the above result by proving that every planar graph with = 6 and without 5-cycles with two chords is of class 1. Recently, Wang and Xu [8] proved that every plane graph G with maximum degree 6 is edge 6-colorable if no vertex in G is incident with four faces of size 3. 2. The main result and its proof To prove our result, we will introduce some known lemmas. Lemma 1. (See [6,9].) If G is a planar graph with (G ) 7, then G is of class 1. Lemma 2. (Vizing’s Adjacency Lemma [2]). Let G be a -critical graph, and let u and v be adjacent vertices of G with d( v ) = k. (a) If k < , then u is adjacent to at least − k + 1 vertices of degree ; (b) If k = , then u is adjacent to at least two vertices of degree . From Vizing’s Adjacency Lemma, it is easy to get the following corollary. Corollary 3. Let G be a -critical graph. Then (a) every vertex is adjacent to at most one 2-vertex and at least two -vertices; (b) the sum of the degree of any two adjacent vertices is at least + 2; (c) if uv ∈ E (G ) and d(u ) + d( v ) = + 2, then every vertex of N ({u , v }) \ {u , v } is a -vertex. Lemma 4. (See [9].) Let G be a -critical graph, uv ∈ E (G ) and d(u ) + d( v ) = + 2. Then (a) every vertex of N ( N ({u , v })) \ {u , v } is of degree at least − 1; (b) if d(u ), d( v ) < , then every vertex of N ( N ({u , v })) \ {u , v } is a -vertex. Lemma 5. (See [6].) No -critical graph has distinct vertices x, y, z such that x is adjacent to y and z, d( z) < 2 − d(x) − d( y ) + 2, and xz is in at least d(x) + d( y ) − − 2 triangles not containing y. To be convenient, we give some definitions and notations on planar graphs. Let G be a plane graph and F (G ) the face set of G. A face of G is said to be incident with all edges and vertices in its boundary. Two faces sharing an edge e are said to be adjacent at e. The degree of a face f of G, denoted by d G ( f ), is the number of edges incident with f where each cut edge is counted twice. A k-, k+ -face is a face of degree k, at least k. A k-face of G is called an (i 1 , i 2 , . . . , ik )-face if the vertices in its boundary are of degrees i 1 , i 2 , . . . , ik respectively. A 3-face is denoted by [x, y , z] if it is incident with distinct vertices x, y , z and d(x) d( y ) d( z). For a vertex v ∈ V (G ), we denote by f k ( v ) the number of k-faces incident with v. Theorem 6. Let G be a planar graph with 6. If any 5-cycle contains at most one chord, then G is of class 1. Proof. Suppose that G is a counterexample to our theorem with the minimum number of edges and suppose that G is embedded in the plane. Then G is a 6-critical graph by Lemma 1, and it is 2-connected and Lemma 2. By Euler’s formula | V (G )| − | E (G )| + | F (G )| = 2, we have
d(x) − 4 +
x∈ V (G )
d(x) − 4 = −8 < 0.
x∈ F (G )
We define ch to be the initial charge. Let ch(x) = d(x) − 4 for each x ∈ V ∪ F . So x∈ V ∪ F ch(x) < 0. In the following, we will reassign a new charge denoted by ch (x) to each x ∈ V ∪ F according to the discharging rules. Since our rules only move charges around, and do not affect the sum. If we can show that ch (x) 0 for each x ∈ V ∪ F , then we get an obvious contradiction 0 x∈ V ∪ F ch (x) = x∈ V ∪ F ch(x) < 0, which completes our proof. A 4-face f = [ w , v , x, y ] is called special if d(x) = 2 and v , x, y form a 3-face. The discharging rules are defined as follows. R1 Let v be a 2-vertex. If v is incident with at least one 5+ -face, then v receives 1 from any of its incident 5+ -face, from each adjacent vertex; Otherwise, if v is incident with a special 4-face f , then v receives adjacent vertex; Otherwise v receives 1 from each adjacent vertex.
1 3
from f ,
5 6
1 2
from each
126
J.-L. Wu, L. Xue / Theoretical Computer Science 518 (2014) 124–127
R2 Every 3-vertex receives
1 3
from each adjacent vertex.
R3 Let f be a 3-face [x, y , z] such that d(x) d( y ) d( z). If 2 d(x) 4 and d( y ) 5, then f receives 1 3
1 2
from y,
1 2
from
z; If d(x) = d( y ) = 4 and d( z) = 6, then f receives 1 from z; If d(x) 5, then f receives from x, y , z, respectively. R4 Let v be a 5-vertex. R4.1 If v is adjacent to a 6-vertex x and incident with a (3, 5, 6)-face [u , v , w ] such that ux ∈ / E (G ) and w = x, then v receives 13 from x; R4.2 Suppose that v is adjacent to a 4-vertex and incident with two adjacent 3-faces uv w and v wx. Then d6 ( v ) 3. If max{d(u ), d(x)} 5, then v receives 13 from w; Otherwise, choose a vertex y ∈ {u , x} with d( y ) = 6. If d( w ) = 4, then v receives 13 from y; Otherwise v receives 16 from y. R5 Let v be a 6-vertex. R5.1 If v is incident with a special 4-face f = [ w , v , x, y ] such that d( y ) = 2, then v sends
1 3
to f ;
R5.2 If v is incident with two 3-faces [u , v , x] and [ v , x, y ] such that d(x) = d( y ) = 4, then v receives
1 6
from u.
Now, let’s begin to check ch (x) 0 for all x ∈ V ∪ F . Let f ∈ F (G ). Then d( f ) 3. If d( f ) 5, then the number of 2-vertices incident with f is at most d( f ) − 4, and it follows that ch ( f ) ch( f ) − (d( f ) − 4) = 0 by R1. Suppose d( f ) = 4. If f is special, then ch ( f ) = 0 + 13 − 13 = 0 by R1 and R5; Otherwise, ch ( f ) = ch( f ) = 0. Suppose d( f ) = 3. Since = 6, f must be the (2+ , 6, 6)-face, (3, 5+ , 6)-face, (4, 4, 6)-face or (4+ , 5, 5)-face by Lemma 2. Hence ch ( f ) = ch( f ) + max{2 × 1 , 1, 3 × 13 } = 0 by R3. 2 Let w ∈ V (G ). Then d( w ) 2. If d( w ) = 2, then w is adjacent to two 6-vertices by Corollary 3, so ch ( w ) = ch( w ) + max{1 + 2 × 12 , 12 + 2 × 56 , 2 × 1} = 0 by R1. If d( w ) = 3, then w is adjacent to three 5+ -vertices by Corollary 3, and it
follows that ch ( w ) = −1 + 3 × 13 = 0 by R2. If d( w ) = 4, then ch ( w ) = ch( w ) = 0. Suppose that d( w ) = 5. We have ch( w ) = 1, f 3 ( w ) 3, min{d(u ) | u ∈ N ( w )} 3, d3 ( w ) 1 and d6 ( w ) 2. Let w 0 , w 1 , . . . , w 4 be neighbors of w and f 0 , f 1 , . . . , f 4 be faces incident with w such that f i is incident with w i and w i +1 , for all i ∈ {0, 1, . . . , 4}, where w 5 = w 0 . If all neighbors of w are 5+ -vertices, then ch ( w ) 1 − 3 × 13 = 0 by R3. Suppose that min0i 4 d( w i ) = 4. If f 3 ( w ) 2, then ch ( w ) 1 − 2 ×
1 2
= 0 by R3; Otherwise, without loss of generality, 1 + 13 ) 2
assume that f 4 , f 0 , f 2 are 3-faces. If d( w 0 ) = 4, then d( w i ) 5 (1 i 4) by Lemma 5. So w sends at most (2 × to its adjacent 3-faces. At the same time, w receives
ch ( w ) 1 +
1 3
− (2 ×
1 2
1 3
from one of its adjacent 6-vertices by R4.2, and it follows that
= 0. The case that d( w 1 ) = 4 or d( w 2 ) = 4 can be similarly checked as above. Suppose that d3 ( w ) = 1, without loss of generality, assume that d( w 1 ) = 3. Then w sends 13 to w 1 by R2 and d6 ( w ) = 4 by Lemma 2. If f 3 ( w ) 1 or w w 1 is not incident with a 3-face, then ch ( w ) 1 − 13 − max{ 12 , 2 × 13 } = 0 by R2 and R3; Otherwise, f 3 ( w ) 2 and w w 1 is incident with a 3-face. Without loss of generality, assume that d( f 1 ) = 3. By R4.1, w 3 , w 4 send 13 to w, respectively. So ch ( w ) 1 + 2 × 13 − 13 − 2 × 12 − 13 = 0 by R2 and R3 (note that f 0 may be also a 3-face). In the following we check the case that d( w ) = 6. Thus we have ch( w ) = 2, f 3 ( w ) 4, d2 ( w ) 1 and d6 ( w ) 2 by Lemma 2. Note that if w is incident with a special 4-face f = [ w , v , x, y ] such that d(x) = 2, then two faces adjacent to f and incident with w are 4+ -faces since G contains no 5-cycles with two chords. We denote by f s4 ( w ) the number of special 4-faces incident with w and f 34 = f 3 ( w ) + f s4 ( w ). So f s4 ( w ) 3, f 34 ( w ) 4 and if f s4 ( w ) 2, then f 3 ( w ) 1. Case 1. w sends
1 3
1 3
+
1 ) 3
to some adjacent 5-vertex v (ref. R4).
Suppose that v is incident with a (3, 5, 6)-face [u , v , x] such that wu ∈ / E (G ) and w = x (see R4.1). Then w may send to v by R4.1. At the same time, w is adjacent to five 6-vertices by Lemma 4, that is, d6 ( w ) = 5. Since f 34 ( w ) 4,
ch ( w ) = 2 − 13 − 4 × 13 > 0. Suppose that d4 ( v ) 1 and v is incident with three 3-faces [ v , v 0 , v 1 ], [ v , v 1 , v 2 ], [ v , v 3 , v 4 ] (see R4.2). Then d6 ( v ) 3. If w = v 1 , then min{d( v 0 ), d( v 2 )} = 4, max{d( v 0 ), d( v 2 )} 5, f 6 ( w ) = 3, and it follows that ch ( w ) 2 − 13 − 2 × 12 − 2 × 13 = 0; Otherwise, w ∈ { v 0 , v 2 } and f 34 ( w ) 3. Since d( v ) = 5, any neighbor of w is the 3+ -vertex and incident with no (3, 5, 6)-faces and (4, 4, 6)-faces. So ch ( w ) 2 − 2 × 16 − 3 × 12 > 0. Case 2. w sends
1 6
to some adjacent 6-vertex v (ref. R5.2).
Then v is incident with two 3-faces [ w , v , x] and [ v , x, y ] such that d(x) = d( y ) = 4. Then f 34 ( w ) 3 and d6 ( w ) = 5 by Lemma 4. So ch ( w ) = 2 − 16 − 12 − 2 × 13 > 0. Case 3. w sends no charge to its adjacent 5+ -vertices. Let k = min{d(u ) | u ∈ N ( w )}. If k 5, then ch ( w ) 2 − 4 × 13 > 0. Suppose that k = 4. Then d6 ( w ) 3 by Lemma 2. If w is incident with two 3-faces [u , w , x] and [ w , x, y ] such that d(x) = d( y ) = 4, then d6 ( w ) = 4, w y is incident with
J.-L. Wu, L. Xue / Theoretical Computer Science 518 (2014) 124–127
a 4+ -face, w receives
1 6
from u, and it follows that ch ( w ) 2 +
1 6
1 2
−2×
1 3
= 0 since f 34 ( w ) 4; Otherwise, = 0. Suppose that k = 3. Then d6 ( w ) 4 by Lemma 2. If d3 ( w ) = 1 and d5+ ( w ) 5, then ch ( w ) 2 − 13 − 2 × ( 12 + 13 ) = 0;
ch ( w ) 2 − max{1 + 2 ×
1 , 4 × 12 } 3
−1−
127
Otherwise, w is incident with two 4− -vertices u, v, then u and v are incident with at most one 3-face by Lemma 5 since d(u ) + d( v ) + d( w ) 3 + 4 + 6 < 14. So f 34 ( w ) 3, and it follows that ch ( w ) 2 − 13 − 12 − 2 × 13 > 0 by R1 and R3. Suppose that k = 2, that is, w is adjacent to a 2-vertex v. Then d6 ( w ) = 5 by Lemma 2. If v is incident with a special 4-face f = [u , v , w , x], then f 3 ( v ) 3 and w sends 56 to v, and it follows that ch ( w ) 2 − 56 − 12 − 2 × 13 = 0; Otherwise, v is incident with a 5+ -face or two 4-faces. If v is incident with a 5+ -face, then w sends 2−
1 2
− ( 12
+3×
1 ) 3
1 2
to v, and it follows that ch ( w )
= 0. If v is incident with two 4-faces, then f 34 3, and it follows that ch ( w ) 2 − 1 − 3 ×
1 3
= 0. 2
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