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First-Fit coloring of {P5,K4−e} -free graphs
Discrete Applied Mathematics 158 (2010) 620–626
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First-Fit coloring of {P5 , K4 − e}-free graphs S.A. Choudum, T. Karthick ∗ Department of Mathematics, Indian Institute of Technology Madras, Chennai - 600 036, India
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Article history: Received 26 December 2008 Received in revised form 3 December 2009 Accepted 17 December 2009 Available online 12 January 2010 Keywords: P5 -free graphs Paw-free graphs On-line coloring First-Fit coloring Clique number
abstract We show that given any ordering of the vertices of a {P5 , K4 − e}-free graph G, the FirstFit coloring algorithm colors its vertices using at most 2ω(G) − 1 colors (where ω(G) is the clique number of G) via a characterization proved by using the known results. We also construct {P5 , K4 − e}-free graphs to show that this bound cannot be improved. A similar result is proved for the class of {P5 , paw}-free graphs. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The class of P5 -free graphs and its subclasses have been studied in a variety of contexts like minimum dominating set problem (MDSP), minimum coloring problem (MCP), maximum independent set problem (MISP) and on-line colorings. It is well known that for P4 -free graphs, all these problems are solvable efficiently [5,8–10]. However, for P5 -free graphs, while MDSP is known to be NP-hard [15], the complexity of MISP is unknown. Thus one is led to study k-colorability problem (k-CP) for the class of Pt -free graphs. For a fixed integer k, the k-colorability problem (k-CP) is to design an algorithm which checks whether a given graph G admits a k-coloring and outputs one if it does. It is well known that k-CP is solvable in O(m)-time for P4 -free graphs; see [18]. However, there remain several open problems on this topic. 3-CP is solvable in polynomial time for P5 -free graphs [20] and P6 -free graphs [19], but the complexity is unknown for Pt (t ≥ 7)-free graphs. 4-CP is NP-complete for Pt (t ≥ 9)-free graphs, see [12], and the complexity is unknown for Pt -free graphs, if t = 6, 7 or 8. k-CP (k ≥ 5) is NP-complete for Pt -free graphs, if t ≥ 8, and the complexity is unknown for P6 -free graphs and P7 -free graphs. An important case was recently settled by Hoang et al. [12] by showing that for every fixed k, k-CP is solvable in polynomial time for P5 -free graphs. These algorithmic complexity issues motivate one to study on-line coloring of Pt -free graphs. In a pioneering paper, Gyárfás and Lehel [10] initiated a systematic study of on-line and First-Fit coloring algorithms of graphs which use no more colors than a function of the clique number. Note that such a function is useful to obtain the approximation factor of an on-line algorithm since the chromatic number of any graph is bounded below by its clique number. We shall call a class G of graphs efficiently on-line colorable, if given any G ∈ G and given any ordering of the vertices of G, there exists an on-line coloring algorithm which uses no more than p(ω(G)) colors, where p is a polynomial. Gyárfás and Lehel [10] showed that split graphs, complements of bipartite graphs, complements of chordal graphs, and P4 -free graphs are efficiently on-line colorable. Subsequent papers on this topic include [7,13]. These aspects motivated research on P5 -free graphs and {P5 , H }-free graphs, for various graphs H; see [1,3,4,6,11,14,16]. In this paper, we are concerned with the First-Fit coloring of two subclasses of P5 -free graphs. We show that given any
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Corresponding author. Fax: +91 044 2257 4602. E-mail addresses: [email protected] (S.A. Choudum), [email protected] (T. Karthick).
0166-218X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2009.12.009
S.A. Choudum, T. Karthick / Discrete Applied Mathematics 158 (2010) 620–626
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ordering of the vertices of a {P5 , K4 − e}-free graph G, the First-Fit coloring algorithm colors its vertices using at most 2ω(G) − 1 colors via a characterization proved by using the known results. We also construct {P5 , K4 − e}-free graphs to show that this bound cannot be improved. A similar result is proved for the class of {P5 , paw}-free graphs. In this respect, the following existing results may be noted.
• Given any positive integer t, Gyárfás and Lehel [10] have constructed a P6 -free bipartite graph G whose vertices can be ordered so that the First-Fit coloring algorithm uses t colors.
• The problem of designing an on-line coloring algorithm which efficiently colors any P5 -free graph is open; see [14]. • Given any ordering of the vertices of a P5 -free graph G, Kierstead et al. [14] have designed an on-line coloring algorithm x which uses at most f (ω(G)) colors, where f (x) = 4 3−1 . • Cieślik [7] has shown that given anyordering of the vertices of a {P5 , C4 }-free graph G, the First-Fit coloring algorithm colors the vertices of G with at most
ω(G)+1 2
colors, and this bound cannot be improved.
2. Notations and terminology All our graphs are finite, undirected and simple. We follow West [21] for general graph theoretic terminology. The symbols Pn , Cn , Kn respectively denote the path, cycle, complete graph on n vertices. The graph K4 − e (called the diamond in several papers) is the complete graph on four vertices minus one edge. Paw is a graph on four vertices a, b, c , d with edges ab, ac , ad, bc. If F is a family of graphs, G is said to be F -free if it contains no induced subgraph isomorphic to any graph in F . If v is a vertex in a graph G, then deg (v) denotes the degree of v and 1(G) := max{deg (v) : v ∈ V (G)}. Also, N (v) denote the set of vertices that are adjacent with v . A clique of a graph G is a complete subgraph of G. The clique number of G is the integer ω(G) = max {|Q |: Q is a clique in G}. If H is an induced subgraph of G, we write H v G. If S and T are two vertex disjoint subsets, then [S , T ] denotes the set of edges with one vertex in S and the other in T . [S , T ] is said to be complete if every vertex in S is adjacent with every vertex in T . [S ] denotes the subgraph induced by S. A set S ⊆ V (G) is called a dominating set of G if every y ∈ V (G) \ S is adjacent to some x ∈ S. If G1 and G2 are two vertex disjoint graphs, then their union G1 ∪ G2 is the graph with V (G1 ∪ G2 ) = V (G1 ) ∪ V (G2 ) and E (G1 ∪ G2 ) = E (G1 ) ∪ E (G2 ). For any positive integer k, kG denotes the union of k graphs each isomorphic with G. Next, we define an operation of combining various graphs. Let G be a graph on n vertices v1 , v2 , . . . , vn , and let H1 , H2 , . . . , Hn be any n vertex disjoint graphs. Then an expansion G(H1 , H2 , . . . , Hn ) of G is the graph obtained from G by (i) replacing the vertex vi of G by Hi , i = 1, 2, . . . , n, and (ii) joining the vertices x ∈ Hi , y ∈ Hj iff vi and vj are adjacent in G. An expansion is also called a composition; see [21]. If every Hi is an edgeless graph, then G(H1 , H2 , . . . , Hn ) is said to be an independent expansion of G, and it is denoted by IG. This operation is called multiplication in [1,21]. 3. Characterization of {P5 , K4 − e}-free graphs We divide the characterization into two parts, the first part containing graphs which do not induce C5 , and the second part containing the graphs which contain an induced C5 . Our aim here is to describe a structure which enables us to derive the least upper bound on the number of colors utilized by the First-Fit coloring algorithm in the worst case. Also, we remark that our characterization of {P5 , K4 − e}-free graphs which do not induce C5 (Theorem 1 below) can also be derived from Theorem 2 of Brandstädt [3] which uses the notion of prime graphs and the expansion of graphs. 3.1. Characterization of {P5 , C5 , K4 − e}-free graphs Our characterization asserts that every connected {P5 , C5 , K4 − e}-free graph belongs to one of the following three classes of graphs. The class K1 : A graph G ∈ K1 if G contains a dominating complete graph D such that (i) every component of G − V (D) is complete, (ii) for each component Q in G − V (D), there is a unique x ∈ D such that [{x}, V (Q )] is complete, and (iv) no other edges in G. The class K2 : A graph G ∈ K2 if G contains a 2K2 -free bipartite graph H with an edge v1 v2 which dominates G such that [V (H )\{v1 , v2 }, V (G)\ V (H )] = ∅ and the components Q1 , Q2 , . . . , Qk of G − V (H ) have the following properties: (i) each Qi is complete, (ii) for each i, either [{v1 }, V (Qi )] is complete or [{v2 }, V (Qi )] is complete, (iii) for at most one i, [{v1 , v2 }, V (Qi )] is complete, and (iv) no other edges in G. The class MC of matched complete graphs: A graph G ∈ MC if V (G) can be partitioned into two sets X and Y such that: (i) [X ], [Y ] are complete, and (ii) [X , Y ] is a non-empty set of independent edges. The following theorem proved by Bacsó and Tuza [2] on dominating cliques in {P5 , C5 }-free graphs is a basic step in our proof. Theorem A ([2]). A connected graph G is {P5 , C5 }-free if and only if every connected induced subgraph of G contains a dominating clique.
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Theorem 1. A connected graph G is {P5 , C5 , K4 − e}-free if and only if G ∈ K1 ∪ K2 ∪ MC. Proof. If G ∈ K1 ∪ K2 ∪ MC, then it is easy to verify that G is {P5 , C5 , K4 − e}-free. To prove the reverse implication we use Theorem A. Among all dominating cliques in G, we choose a dominating clique of maximum size (say) D = [{v1 , v2 , . . . , vk }]. Since D is maximum and G is (K4 − e)-free, any x ∈ V (G) \ V (D) is adjacent to exactly one vertex of D. So, we define Vi = {x ∈ V (G) \ V (D) : N (x) ∩ D = {vi }},
1 ≤ i ≤ k.
Then, since G is (K4 − e)-free, [Vi ] is P3 -free, and so it is a union of (disjoint) complete subgraphs of G, for all i, 1 ≤ i ≤ k. If [Vi , Vj ] = ∅, for all i and j(i 6= j), then G ∈ K1 . So, assume that [Vi , Vj ] 6= ∅, for some i and j(i 6= j). W. l. o. g., let [V1 , V2 ] 6= ∅ and let xy ∈ [V1 , V2 ]. Case 1: k ≥ 3. We first prove the following assertions. (1) xz , yz ∈ E (G), for all z ∈ i=3 Vi . For otherwise, there exist i and z such that z ∈ Vi , and xz 6∈ E (G) or yz 6∈ E (G). Consequently, we have the following contradictions: (a) If xz 6∈ E (G) and yz 6∈ E (G), then [{z , vi , v2 , y, x}] ∼ = P5 . (b) If xz ∈ E (G) and yz 6∈ E (G), then [{x, y, v2 , vi , z }] ∼ = C5 . (c) If xz 6∈ E (G) and yz ∈ E (G), then [{v1 , x, y, z , vi }] ∼ = C5 .
Sk
(2) [ i=3 Vi ] is complete; else [{x, y, z , z1 }] ∼ = K4 − e, where z , z1 are two nonadjacent vertices in [ (3) |Vi | ≤ 1, for every i, 3 ≤ i ≤ k; else [{x, z , z1 , vi }] ∼ = K4 − e, where z , z1 ∈ Vi .
Sk
Sk
i=3
Vi ].
Case 1.1: Vi 6= ∅, for some i (3 ≤ i ≤ k). (In this case, we show that G ∈ MC). W. l. o. g., assume that V3 6= ∅ and let z ∈ V3 . (4) |V1 | = 1 (Similarly, |V2 | = 1). On the contrary, assume that there exists x1 (6= x) ∈ V1 . We deduce the following contradictions: (a) If xx1 ∈ E (G), then zx1 ∈ E (G) (else, [{x1 , x, z , v3 , v2 }] ∼ = P5 ). But, then [{z , x, x1 , v1 }] ∼ = K4 − e. (b) If xx1 6∈ E (G) and zx1 6∈ E (G), then [{x, v1 , v3 , z , y}] ∼ = P5 or C5 according as yx1 6∈ E (G) or yx1 ∈ E (G) respectively. (c) If xx1 ∈ 6 E (G) and zx1 ∈ E (G), then x1 y 6∈ E (G) (else, [{x, v1 , y, z }] ∼ = K4 − e). But, then [{x, v1 , v2 , y, z }] ∼ = C5 . Hence, assertions (1), (2), (3) and (4) imply that G ∈ MC[D, V (G) \ D].
Sk
Case 1.2: i=3 Vi = ∅. (In this case, we show that G ∈ K2 ). Let X = {x ∈ V1 : N (x) ∩ V2 6= ∅} and Y = {y ∈ V2 : N (y) ∩ V1 6= ∅}. Then it is easy to see that [X ] and [Y ] are independent (else, P5 or K4 − e v G). Let H = [X ∪ Y ∪ {v1 } ∪ {v2 }]. Then H is a 2K2 -free bipartite graph (else, P5 v G) with an edge v1 v2 which dominates G. Moreover, G − V (H ) is a disjoint union of complete subgraphs of G. Hence, G ∈ K2 . Case 2: k = 2. Case 2.1: xx1 ∈ E (G), for some x1 ∈ V1 . (In this case, we show that G ∈ MC). If every vertex in V2 is adjacent to either x or x1 , then {x, x1 , v1 } is a dominating clique in G of size 3, a contradiction to the maximality of k. So, there exists a vertex w ∈ V2 such that w x, w x1 6∈ E (G). If w y 6∈ E (G), then since yx1 6∈ E (G), [{w, v2 , y, x, x1 }] is an induced P5 . So, wy ∈ E (G). Hence, if W = {w ∈ V2 : w x, wx1 6∈ E (G)}, then [W ∪ {y}] is a clique in G, since [V2 ] is P3 -free. Next, consider V2 \ W = {z ∈ V2 : zx or zx1 ∈ E (G)}. Then we claim that every z ∈ V2 \ W is adjacent with every w ∈ W . For otherwise, there exist nonadjacent vertices z , w , where z ∈ V2 \ W and w ∈ W . Since z ∈ V2 \ W , we can assume (w. l. o. g.,) that zx ∈ E (G). But then [{x1 , x, z , v2 , w}] is an induced P5 . Hence the claim holds. It follows that [V2 ] is a clique, since [V2 ] is P3 -free. Similarly, [V1 ] is a clique, since yw ∈ E ([V2 ]). Also, [V1 ∪ {v1 }, V2 ∪ {v2 }] is a set of independent edges in G (else, K4 − e v G). Hence, we conclude that G ∈ MC. Similarly, G ∈ MC if yy1 ∈ E (G), for some y1 ∈ V2 . Case 2.2: xx1 6∈ E (G), for all x1 ∈ V1 , and yy1 6∈ E (G), for all y1 ∈ V2 , that is, deg[V1 ] (x) = deg[V2 ] (y) = 0. (In this case, we show that G ∈ K2 ). Let X = {v ∈ V1 : deg[V1 ] (v) = 0}, Y = {v ∈ V2 : deg[V2 ] (v) = 0} and H = [X ∪ Y ∪ {v1 } ∪ {v2 }]. Then H is a 2K2 -free bipartite graph (else, P5 v G) with an edge v1 v2 which dominates G, and [(V1 \ X ) ∪ (V2 \ Y )] is a disjoint union of complete subgraphs of G. Therefore, G ∈ K2 . 3.2. Characterization of {P5 , K4 − e}-free graphs that contain an induced C5 In this section, we derive a characterization of {P5 , K4 − e}-free graphs that contain an induced C5 using a result proved by Arbib and Mosca [1]. The eleven graphs shown in Fig. 1 are {P5 , K4 − e}-free and contain an induced C5 (shown in bold lines). Let Dk denote the set of vertices at distance k from this C5 .
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Fig. 1. Basic graphs.
Theorem B (Theorem 1.6 of [1]). Any connected {P5 , K4 − e}-free graph that properly contains an induced C5 can be obtained from one of the graphs shown in Fig. 1 by expanding some v ∈ D1 ∪ C5 by independent sets and/or expanding some v ∈ D2 by a P3 -free graph. Note that in Figure 3 of [1], the edge a0 b0 is missing in G4 , and G6 is isomorphic to G7 . So, for convenience, we have renumbered the figures. More importantly note that every graph obtained as described in Theorem B is not a {P5 , K4 − e}free graph. For example, by expanding the vertex p of G6 (see Fig. 1) by an independent set of size ≥ 2, we get a graph which contains an induced K4 − e. In this respect, the following easy observation is useful to obtain the converse of Theorem B. Observation 1. Let G0 be a connected {P5 , K4 − e}-free graph on vertices v1 , v2 , . . . , vn that contains an induced C5 and let G = G0 (H1 , H2 , . . . , Hn ) be an expansion of G0 . Then G is a {P5 , K4 − e}-free graph that contains an induced C5 if and only if the expansion is obtained as follows. For any i, 1 ≤ i ≤ n:
• If vi has exactly one neighbor in G0 , then Hi is P3 -free. • If vi has two nonadjacent neighbors and has no two adjacent neighbors in G0 , then Hi is an independent set. • If vi has two nonadjacent neighbors and two adjacent neighbors in G0 , then Hi is K1 . So, combining Theorem B and Observation 1, we deduce the following result. Theorem 2. Let G be a connected graph. Then G is {P5 , K4 − e}-free and contains an induced C5 if and only if G is obtained from one of the basic graphs shown in Fig. 1 by expanding (i) each vertex shown under a circle by an independent set, and (ii) the vertex shown under a square by a P3 -free graph (see Fig. 1). We thank one of the referees who indicated Observation 1 and outlined its use in the proof of Theorem 2. This considerably shortened our original proof. 4. First-Fit coloring of graphs The First-Fit coloring algorithm (FFC) receives the vertices of a graph G in some order (v1 , v2 , . . . ). At any instant i, it performs two operations: (i) Receives vi and all the edges that vi is incident with v1 , v2 , . . . or vi−1 (but receives no information on which vertex succeeds vi ). (ii) Colors vi with the first available color (∈ {1, 2, 3, . . .}) that does not appear on any vertex adjacent with vi (and subsequently does not change this color). The number of colors used by the FFC depends on the order of the vertices it receives. The parameter χFF (G) denotes the maximum number of colors used by the FFC among all the orderings of the vertices of G. An on-line coloring algorithm (OLC) performs exactly as above except that while coloring vi in Step (ii), it need not select the first available color. Hence, FFC is an OLC, but an OLC need not be FFC. Given an ordering of the vertices of G, an OLC may use more than, less than or same number of colors as that of FFC. For example, the vertices of C5 (K2 , K2 , K2 , K2 , K2 ) can be ordered so that FFC performs poorer than OLC. For such comparisons, and for a survey on OLC and FFC; see [10,13]. Clearly,
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• The FFC of G induces a partition V (G) = (V1 , V2 , . . . , Vk ) such that (i) [Vi ] is independent, for all i (1 ≤ i ≤ k), and (ii) every vertex in Vj is adjacent to some vertex in Vi , for every i, j where 1 ≤ i < j ≤ k. • χFF (G) ≤ 1(G) + 1. • χFF (H ) ≤ χFF (G), for every H v G. If x ∈ V (G), let cFF (x) denote the color of x in a FFC of G. For S ⊆ V (G), define CFF (S ) := {cFF (v) : v is a vertex in [S ]}. We use often the following crucial observation which is implicit in the proof of Theorem 2.5 of [7]. Lemma 1 ([7]). Let S be any vertex subset of G. If [P ] and [Q ] are any two components of G − S, then CFF (S ∪ P ) ⊆ CFF (S ∪ Q ) or CFF (S ∪ Q ) ⊆ CFF (S ∪ P ). So, there exists a component D of G − S such that CFF (V (G)) ⊆ CFF (S ∪ D). An easy consequence of Lemma 1 by setting S = ∅ is the following. Lemma 2. For any graph G, χFF (G) = max{χFF (D) : D is a component of G}.
Next, we prove a lemma which enables us to relate χFF (G) and χFF (IG). Lemma 3. If u and v are two nonadjacent vertices in a graph G such that N (u) = N (v), then (i) cFF (u) = cFF (v), and (ii) χFF (G) = χFF (G − u). Proof. (i) Let χFF (G) = k and let (V1 , V2 , . . . , Vk ) be the partition of V (G) induced by the FFC. The assertion (i) is proved if we show that u, v ∈ Vi , for some i. Assume that u ∈ Vi and v ∈ Vj , where i < j. Since uv 6∈ E (G), FFC ensures that there exists a vertex x ∈ Vi such that xv ∈ E (G). Since N (u) = N (v), xu ∈ E (G), which contradicts that Vi is an independent set. (ii) Follows by (i). We also use the following theorem proved by Gyárfás and Lehel [11]. Theorem C ([11]). If G is a {P5 , K3 }-free graph, then χFF (G) ≤ 3.
4.1. First-Fit coloring of {P5 , K4 − e}-free graphs Theorem 3. If G is a {P5 , K4 − e}-free graph, then χFF (G) ≤ 2ω(G) − 1. Proof. Consider a FFC of G which uses χFF (G) colors and let (V1 , V2 , V3 , . . . , VχFF (G) ) be the consequent partition of V (G); so χFF (G) = |CFF (V (G))|. In view of Lemma 2, it is enough if we prove the theorem for connected graphs with clique number at least 2. If ω(G) = 2, then G is a {P5 , K3 }-free graph, and the result follows by Theorem C. In the following, all our graphs have ω(G) ≥ 3. (a) Assume that G is C5 -free. By Theorem 1, G ∈ K1 ∪ K2 ∪ MC. We follow the notation which we used in the definition of K1 , K2 and MC. (i) If G ∈ K1 , then by Lemma 1, there exists a complete graph Q in G − V (D) such that CFF (V (G)) ⊆ CFF (V (D) ∪ V (Q )). Since |CFF (V (D) ∪ V (Q ))| ≤ |V (D) ∪ V (Q )| ≤ 2ω(G) − 1, we have χFF (G) ≤ 2ω(G) − 1. (ii) If G ∈ K2 , then by Lemma 1, there exists a component P in G − {v1 , v2 } such that CFF (V (G)) ⊆ CFF (V (P ) ∪ {v1 , v2 }), where P is either a complete graph or a 2K2 -free bipartite graph. If P is a complete graph, then |CFF (V (G))| ≤ |CFF (V (P ) ∪ {v1 , v2 })| ≤ (ω(G) − 1) + 2 = ω(G) + 1 ≤ 2ω(G) − 1. If P is a 2K2 -free bipartite graph, we assume that |CFF (V (P ) ∪ {v1 , v2 })| ≥ 2ω(G), and arrive at a contradiction. Since ω(G) ≥ 3 and CFF (V (G)) ⊆ CFF (V (P ) ∪ {v1 , v2 }), there are at least six colors in CFF (V (P ) ∪ {v1 , v2 }). Let i, j, k and l be the colors in CFF (V (P )) \ CFF ({v1 , v2 }). Therefore, cFF (v1 ) 6= i, j, k, l and cFF (v2 ) 6= i, j, k, l. Let Ut = Vt ∩ V (P ), for t ∈ {i, j, k, l}. W. l. o. g., assume that i < j < k < l. Let x ∈ Uj . Then since j > i, by the FFC principle, there exist y ∈ Ui such that xy ∈ E (P ). Claim: If a ∈ Uk or Ul , then ax ∈ E (P ). On the contrary, assume that ax 6∈ E (P ). Then, there exists x0 ∈ Uj such that ax0 ∈ E (P ). Now, (1) if ay ∈ E (P ), then 0 x y 6∈ E (P ) (else, K3 v P). So, there exists y0 ∈ Ui such that x0 y0 ∈ E (P ). Then y0 a 6∈ E (P ) (else, K3 v P). Since P is 2K2 free, y0 x ∈ E (P ). But, then [{a, y, x, y0 , x0 }] ∼ = C5 v P, a contradiction. Next, (2) if ay 6∈ E (P ), then there exist y0 ∈ Ui and x0 ∈ Uj such that ay0 , ax0 ∈ E (P ). Since P is a 2K2 -free bipartite graph, xy0 , x0 y ∈ E (P ) and x0 y0 6∈ E (P ). But, then [{a, y0 , x, y, x0 }] ∼ = C5 v P, a contradiction. So, ax ∈ E (P ). Hence the claim holds. Let z ∈ Uk and t ∈ Ul . Then by above claim, zx, tx ∈ E (P ). Then tz 6∈ E (P ) (else, K3 v P). So, there exists z 0 ∈ Uk such that tz 0 ∈ E (P ). Then z 0 x 6∈ E (P ) (else, [{z 0 , x, t }] ∼ = K3 v P). So, there exists x0 ∈ Uj such that z 0 x0 ∈ E (P ). Since P is 2K2 -free, x0 z ∈ E (P ) and since it is bipartite x0 t 6∈ E (P ). But, then [{t , x, z , x0 , z 0 }] ∼ = C5 v P, a contradiction. Hence, |CFF (V (P ) ∪ {v1 , v2 })| ≤ 2ω(G) − 1, and so χFF (G) = |CFF (V (G))| ≤ |CFF (V (P ) ∪ {v1 , v2 })| ≤ 2ω(G) − 1. (iii) If G ∈ MC, then it is easy to see that deg (v) ≤ ω(G), for all v ∈ V (G). Hence, χFF (G) ≤ 1(G)+ 1 ≤ ω(G)+ 1 ≤ 2ω(G)− 1. (b) Assume that G contains an induced C5 . (In this case, we use Theorem 2.)
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Fig. 2. The vertex indicated in square is expanded by a P3 -free graph.
By Theorem 2, G is obtained from one of eleven basic graphs H as described in Theorem 2. If H 6= G10 , then by repeatedly applying Lemma 3(ii) to G, we have χFF (G) = χFF (H ). It can be easily verified that χFF (H ) ≤ 2ω(H ) − 1. Also, ω(G) = ω(H ), since the independent expansion does not change the clique number. Hence, χFF (G) ≤ 2ω(G) − 1. If H = G10 , then by repeatedly applying Lemma 3(ii) to G, we have χFF (G) = χFF (G∗ ), where G∗ is the graph shown in Fig. 2. Also, ω(G) = ω(G∗ ) as above. By Lemma 1, there exists a component D of G∗ − S, where S = {x, y, z }, such that CFF (V (G∗ )) ⊆ CFF (S ∪ V (D)). Since D is complete and |CFF (S ∪ V (D))| ≤ |S ∪ V (D)| ≤ 3 + (ω(G∗ ) − 1) ≤ 2ω(G∗ ) − 1, we have χFF (G) = χFF (G∗ ) ≤ 2ω(G∗ ) − 1 = 2ω(G) − 1. Remark. The bound given in Theorem 3 cannot be improved. To justify this claim, we construct a {P5 , K4 − e}-free graph G with ω(G) = t and present an ordering of its vertices for which the First-Fit coloring uses 2t − 1 colors.
• We first present ‘t’ vertex disjoint copies of Kt −1 , say Kt1−1 , Kt2−1 , . . . , Ktt−1 . The FFC assigns the colors 1, 2, 3, . . . , t − 1 to the vertices of Kti−1 (1 ≤ i ≤ t). • Next, we present a complete subgraph K ∼ = [{v1 , v2 , v3 , . . . , vt }] such that for each i (1 ≤ i ≤ t), (1) V (K ) ∩ V (Kti−1 ) = ∅, j i (2) [{vi }∪V (Kt −1 )] is a complete subgraph and (3) [{vi }, V (Kt −1 )] = ∅, for all j 6= i. Now, v1 , v2 , v3 , . . . , vt are respectively colored by t , t + 1, t + 2, . . . , 2t − 1 by the FFC. The graph G so constructed belongs to K1 , and hence it is {P5 , C5 , K4 − e}-free and ω(G) = t. 4.2. First-Fit coloring of {P5 , paw}-free graphs A characterization of paw -free graphs by Olariu [17] is the following. Theorem D ([17]). Let G be a connected graph. Then G is paw -free if and only if G is triangle-free or complete multipartite.
Theorem 4. If G is a {P5 , paw}-free graph, then χFF (G) ≤ ω(G) + 1. Proof. In view of Lemma 2, it is enough if we prove the theorem for connected graphs with clique number at least 2. If G is triangle-free, then the result follows by Theorem C. If G is a complete multipartite graph, then it is an independent expansion of a complete graph Q . So, by repeatedly applying Lemma 3(ii) to G, we have χFF (G) = χFF (Q ) = |Q | = ω(G). Given any ordering of the vertices of the {P5 , K3 }-free graph C5 (Kmc , Kmc , Kmc , Kmc , Kmc ), m ≥ 1, FFC uses three colors. Hence the upper bound given in Theorem 4 cannot be improved. Acknowledgements The authors gratefully acknowledge the referees for their suggestions and comments which improved the presentation of the paper. The second author acknowledges CSIR, India for the financial support to carry out this research work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
C. Arbib, R. Mosca, On (P5 , diamond)-free graphs, Discrete Math. 250 (2002) 1–22. G. Bacsó, Z. Tuza, Dominating cliques in P5 -free graphs, Period. Math. Hungar. 21 (4) (1990) 303–308. A. Brandstädt, (P5 , diamond)-free graphs revisited: Structure and linear time optimization, Discrete Appl. Math. 138 (2004) 13–27. A. Brandstädt, R. Mosca, On the structure and stability number of P5 - and co-chair-free graphs, Discrete Appl. Math. 132 (2004) 47–65. H.S. Chao, F.R. Hsu, R.C.T. Lee, An optimal algorithm for finding the minimum cardinality dominating set on permutation graphs, Discrete Appl. Math. 102 (3) (2000) 159–173. S.A. Choudum, T. Karthick, M.A. Shalu, Perfect coloring and linearly χ -bound P6 -free graphs, J. Graph Theory 54 (4) (2007) 293–306. I. Cieślik, On-line coloring of P5 -free graphs, Acta Inform. 45 (2008) 79–91. D.G. Corneil, H. Lerchs, L. Burlingham, L.K. Stewart, Complement reducible graphs, Discrete Appl. Math. 3 (1981) 163–174. V. Giakoumakis, F. Roussel, H. Thuillier, On P4 -tidy graphs, Discrete Math. Theor. Comput. Sci. 1 (1997) 17–41. A. Gyárfás, J. Lehel, On-line and first fit colorings of graphs, J. Graph Theory 12 (2) (1988) 217–227. A. Gyárfás, J. Lehel, Effective on-line coloring of P5 -free graphs, Combinatorica 11 (2) (1991) 181–184. C.T. Hoang, M. Kamiński, V. Lozin, J. Sawada, X. Shu, Deciding k-colorability of P5 -free graphs in polynomial time, Algorithmica, in press (doi:10.1007/s00453-008-9197-8). H.A. Kierstead, Coloring graphs on-line, in: On-line Algorithms: The State of the Art, in: LNCS, vol. 1442, 1998, pp. 281–305. H.A. Kierstead, S.G. Penrice, W.T. Trotter, On-line first fit coloring of graphs that do not induce P5 , SIAM J. Discrete Math. 8 (4) (1995) 485–498. D.V. Korobitsyn, On the complexitity of determining the domination number in monogenic classes of graphs, Discrete Math. Appl. 2 (2) (1992) 191–199.
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First-Fit coloring of {P5 , K4 − e}-free graphs S.A. Choudum, T. Karthick ∗ Department of Mathematics, Indian Institute of Technology Madras, Chennai - 600 036, India
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Article history: Received 26 December 2008 Received in revised form 3 December 2009 Accepted 17 December 2009 Available online 12 January 2010 Keywords: P5 -free graphs Paw-free graphs On-line coloring First-Fit coloring Clique number
abstract We show that given any ordering of the vertices of a {P5 , K4 − e}-free graph G, the FirstFit coloring algorithm colors its vertices using at most 2ω(G) − 1 colors (where ω(G) is the clique number of G) via a characterization proved by using the known results. We also construct {P5 , K4 − e}-free graphs to show that this bound cannot be improved. A similar result is proved for the class of {P5 , paw}-free graphs. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The class of P5 -free graphs and its subclasses have been studied in a variety of contexts like minimum dominating set problem (MDSP), minimum coloring problem (MCP), maximum independent set problem (MISP) and on-line colorings. It is well known that for P4 -free graphs, all these problems are solvable efficiently [5,8–10]. However, for P5 -free graphs, while MDSP is known to be NP-hard [15], the complexity of MISP is unknown. Thus one is led to study k-colorability problem (k-CP) for the class of Pt -free graphs. For a fixed integer k, the k-colorability problem (k-CP) is to design an algorithm which checks whether a given graph G admits a k-coloring and outputs one if it does. It is well known that k-CP is solvable in O(m)-time for P4 -free graphs; see [18]. However, there remain several open problems on this topic. 3-CP is solvable in polynomial time for P5 -free graphs [20] and P6 -free graphs [19], but the complexity is unknown for Pt (t ≥ 7)-free graphs. 4-CP is NP-complete for Pt (t ≥ 9)-free graphs, see [12], and the complexity is unknown for Pt -free graphs, if t = 6, 7 or 8. k-CP (k ≥ 5) is NP-complete for Pt -free graphs, if t ≥ 8, and the complexity is unknown for P6 -free graphs and P7 -free graphs. An important case was recently settled by Hoang et al. [12] by showing that for every fixed k, k-CP is solvable in polynomial time for P5 -free graphs. These algorithmic complexity issues motivate one to study on-line coloring of Pt -free graphs. In a pioneering paper, Gyárfás and Lehel [10] initiated a systematic study of on-line and First-Fit coloring algorithms of graphs which use no more colors than a function of the clique number. Note that such a function is useful to obtain the approximation factor of an on-line algorithm since the chromatic number of any graph is bounded below by its clique number. We shall call a class G of graphs efficiently on-line colorable, if given any G ∈ G and given any ordering of the vertices of G, there exists an on-line coloring algorithm which uses no more than p(ω(G)) colors, where p is a polynomial. Gyárfás and Lehel [10] showed that split graphs, complements of bipartite graphs, complements of chordal graphs, and P4 -free graphs are efficiently on-line colorable. Subsequent papers on this topic include [7,13]. These aspects motivated research on P5 -free graphs and {P5 , H }-free graphs, for various graphs H; see [1,3,4,6,11,14,16]. In this paper, we are concerned with the First-Fit coloring of two subclasses of P5 -free graphs. We show that given any
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Corresponding author. Fax: +91 044 2257 4602. E-mail addresses: [email protected] (S.A. Choudum), [email protected] (T. Karthick).
0166-218X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2009.12.009
S.A. Choudum, T. Karthick / Discrete Applied Mathematics 158 (2010) 620–626
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ordering of the vertices of a {P5 , K4 − e}-free graph G, the First-Fit coloring algorithm colors its vertices using at most 2ω(G) − 1 colors via a characterization proved by using the known results. We also construct {P5 , K4 − e}-free graphs to show that this bound cannot be improved. A similar result is proved for the class of {P5 , paw}-free graphs. In this respect, the following existing results may be noted.
• Given any positive integer t, Gyárfás and Lehel [10] have constructed a P6 -free bipartite graph G whose vertices can be ordered so that the First-Fit coloring algorithm uses t colors.
• The problem of designing an on-line coloring algorithm which efficiently colors any P5 -free graph is open; see [14]. • Given any ordering of the vertices of a P5 -free graph G, Kierstead et al. [14] have designed an on-line coloring algorithm x which uses at most f (ω(G)) colors, where f (x) = 4 3−1 . • Cieślik [7] has shown that given anyordering of the vertices of a {P5 , C4 }-free graph G, the First-Fit coloring algorithm colors the vertices of G with at most
ω(G)+1 2
colors, and this bound cannot be improved.
2. Notations and terminology All our graphs are finite, undirected and simple. We follow West [21] for general graph theoretic terminology. The symbols Pn , Cn , Kn respectively denote the path, cycle, complete graph on n vertices. The graph K4 − e (called the diamond in several papers) is the complete graph on four vertices minus one edge. Paw is a graph on four vertices a, b, c , d with edges ab, ac , ad, bc. If F is a family of graphs, G is said to be F -free if it contains no induced subgraph isomorphic to any graph in F . If v is a vertex in a graph G, then deg (v) denotes the degree of v and 1(G) := max{deg (v) : v ∈ V (G)}. Also, N (v) denote the set of vertices that are adjacent with v . A clique of a graph G is a complete subgraph of G. The clique number of G is the integer ω(G) = max {|Q |: Q is a clique in G}. If H is an induced subgraph of G, we write H v G. If S and T are two vertex disjoint subsets, then [S , T ] denotes the set of edges with one vertex in S and the other in T . [S , T ] is said to be complete if every vertex in S is adjacent with every vertex in T . [S ] denotes the subgraph induced by S. A set S ⊆ V (G) is called a dominating set of G if every y ∈ V (G) \ S is adjacent to some x ∈ S. If G1 and G2 are two vertex disjoint graphs, then their union G1 ∪ G2 is the graph with V (G1 ∪ G2 ) = V (G1 ) ∪ V (G2 ) and E (G1 ∪ G2 ) = E (G1 ) ∪ E (G2 ). For any positive integer k, kG denotes the union of k graphs each isomorphic with G. Next, we define an operation of combining various graphs. Let G be a graph on n vertices v1 , v2 , . . . , vn , and let H1 , H2 , . . . , Hn be any n vertex disjoint graphs. Then an expansion G(H1 , H2 , . . . , Hn ) of G is the graph obtained from G by (i) replacing the vertex vi of G by Hi , i = 1, 2, . . . , n, and (ii) joining the vertices x ∈ Hi , y ∈ Hj iff vi and vj are adjacent in G. An expansion is also called a composition; see [21]. If every Hi is an edgeless graph, then G(H1 , H2 , . . . , Hn ) is said to be an independent expansion of G, and it is denoted by IG. This operation is called multiplication in [1,21]. 3. Characterization of {P5 , K4 − e}-free graphs We divide the characterization into two parts, the first part containing graphs which do not induce C5 , and the second part containing the graphs which contain an induced C5 . Our aim here is to describe a structure which enables us to derive the least upper bound on the number of colors utilized by the First-Fit coloring algorithm in the worst case. Also, we remark that our characterization of {P5 , K4 − e}-free graphs which do not induce C5 (Theorem 1 below) can also be derived from Theorem 2 of Brandstädt [3] which uses the notion of prime graphs and the expansion of graphs. 3.1. Characterization of {P5 , C5 , K4 − e}-free graphs Our characterization asserts that every connected {P5 , C5 , K4 − e}-free graph belongs to one of the following three classes of graphs. The class K1 : A graph G ∈ K1 if G contains a dominating complete graph D such that (i) every component of G − V (D) is complete, (ii) for each component Q in G − V (D), there is a unique x ∈ D such that [{x}, V (Q )] is complete, and (iv) no other edges in G. The class K2 : A graph G ∈ K2 if G contains a 2K2 -free bipartite graph H with an edge v1 v2 which dominates G such that [V (H )\{v1 , v2 }, V (G)\ V (H )] = ∅ and the components Q1 , Q2 , . . . , Qk of G − V (H ) have the following properties: (i) each Qi is complete, (ii) for each i, either [{v1 }, V (Qi )] is complete or [{v2 }, V (Qi )] is complete, (iii) for at most one i, [{v1 , v2 }, V (Qi )] is complete, and (iv) no other edges in G. The class MC of matched complete graphs: A graph G ∈ MC if V (G) can be partitioned into two sets X and Y such that: (i) [X ], [Y ] are complete, and (ii) [X , Y ] is a non-empty set of independent edges. The following theorem proved by Bacsó and Tuza [2] on dominating cliques in {P5 , C5 }-free graphs is a basic step in our proof. Theorem A ([2]). A connected graph G is {P5 , C5 }-free if and only if every connected induced subgraph of G contains a dominating clique.
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Theorem 1. A connected graph G is {P5 , C5 , K4 − e}-free if and only if G ∈ K1 ∪ K2 ∪ MC. Proof. If G ∈ K1 ∪ K2 ∪ MC, then it is easy to verify that G is {P5 , C5 , K4 − e}-free. To prove the reverse implication we use Theorem A. Among all dominating cliques in G, we choose a dominating clique of maximum size (say) D = [{v1 , v2 , . . . , vk }]. Since D is maximum and G is (K4 − e)-free, any x ∈ V (G) \ V (D) is adjacent to exactly one vertex of D. So, we define Vi = {x ∈ V (G) \ V (D) : N (x) ∩ D = {vi }},
1 ≤ i ≤ k.
Then, since G is (K4 − e)-free, [Vi ] is P3 -free, and so it is a union of (disjoint) complete subgraphs of G, for all i, 1 ≤ i ≤ k. If [Vi , Vj ] = ∅, for all i and j(i 6= j), then G ∈ K1 . So, assume that [Vi , Vj ] 6= ∅, for some i and j(i 6= j). W. l. o. g., let [V1 , V2 ] 6= ∅ and let xy ∈ [V1 , V2 ]. Case 1: k ≥ 3. We first prove the following assertions. (1) xz , yz ∈ E (G), for all z ∈ i=3 Vi . For otherwise, there exist i and z such that z ∈ Vi , and xz 6∈ E (G) or yz 6∈ E (G). Consequently, we have the following contradictions: (a) If xz 6∈ E (G) and yz 6∈ E (G), then [{z , vi , v2 , y, x}] ∼ = P5 . (b) If xz ∈ E (G) and yz 6∈ E (G), then [{x, y, v2 , vi , z }] ∼ = C5 . (c) If xz 6∈ E (G) and yz ∈ E (G), then [{v1 , x, y, z , vi }] ∼ = C5 .
Sk
(2) [ i=3 Vi ] is complete; else [{x, y, z , z1 }] ∼ = K4 − e, where z , z1 are two nonadjacent vertices in [ (3) |Vi | ≤ 1, for every i, 3 ≤ i ≤ k; else [{x, z , z1 , vi }] ∼ = K4 − e, where z , z1 ∈ Vi .
Sk
Sk
i=3
Vi ].
Case 1.1: Vi 6= ∅, for some i (3 ≤ i ≤ k). (In this case, we show that G ∈ MC). W. l. o. g., assume that V3 6= ∅ and let z ∈ V3 . (4) |V1 | = 1 (Similarly, |V2 | = 1). On the contrary, assume that there exists x1 (6= x) ∈ V1 . We deduce the following contradictions: (a) If xx1 ∈ E (G), then zx1 ∈ E (G) (else, [{x1 , x, z , v3 , v2 }] ∼ = P5 ). But, then [{z , x, x1 , v1 }] ∼ = K4 − e. (b) If xx1 6∈ E (G) and zx1 6∈ E (G), then [{x, v1 , v3 , z , y}] ∼ = P5 or C5 according as yx1 6∈ E (G) or yx1 ∈ E (G) respectively. (c) If xx1 ∈ 6 E (G) and zx1 ∈ E (G), then x1 y 6∈ E (G) (else, [{x, v1 , y, z }] ∼ = K4 − e). But, then [{x, v1 , v2 , y, z }] ∼ = C5 . Hence, assertions (1), (2), (3) and (4) imply that G ∈ MC[D, V (G) \ D].
Sk
Case 1.2: i=3 Vi = ∅. (In this case, we show that G ∈ K2 ). Let X = {x ∈ V1 : N (x) ∩ V2 6= ∅} and Y = {y ∈ V2 : N (y) ∩ V1 6= ∅}. Then it is easy to see that [X ] and [Y ] are independent (else, P5 or K4 − e v G). Let H = [X ∪ Y ∪ {v1 } ∪ {v2 }]. Then H is a 2K2 -free bipartite graph (else, P5 v G) with an edge v1 v2 which dominates G. Moreover, G − V (H ) is a disjoint union of complete subgraphs of G. Hence, G ∈ K2 . Case 2: k = 2. Case 2.1: xx1 ∈ E (G), for some x1 ∈ V1 . (In this case, we show that G ∈ MC). If every vertex in V2 is adjacent to either x or x1 , then {x, x1 , v1 } is a dominating clique in G of size 3, a contradiction to the maximality of k. So, there exists a vertex w ∈ V2 such that w x, w x1 6∈ E (G). If w y 6∈ E (G), then since yx1 6∈ E (G), [{w, v2 , y, x, x1 }] is an induced P5 . So, wy ∈ E (G). Hence, if W = {w ∈ V2 : w x, wx1 6∈ E (G)}, then [W ∪ {y}] is a clique in G, since [V2 ] is P3 -free. Next, consider V2 \ W = {z ∈ V2 : zx or zx1 ∈ E (G)}. Then we claim that every z ∈ V2 \ W is adjacent with every w ∈ W . For otherwise, there exist nonadjacent vertices z , w , where z ∈ V2 \ W and w ∈ W . Since z ∈ V2 \ W , we can assume (w. l. o. g.,) that zx ∈ E (G). But then [{x1 , x, z , v2 , w}] is an induced P5 . Hence the claim holds. It follows that [V2 ] is a clique, since [V2 ] is P3 -free. Similarly, [V1 ] is a clique, since yw ∈ E ([V2 ]). Also, [V1 ∪ {v1 }, V2 ∪ {v2 }] is a set of independent edges in G (else, K4 − e v G). Hence, we conclude that G ∈ MC. Similarly, G ∈ MC if yy1 ∈ E (G), for some y1 ∈ V2 . Case 2.2: xx1 6∈ E (G), for all x1 ∈ V1 , and yy1 6∈ E (G), for all y1 ∈ V2 , that is, deg[V1 ] (x) = deg[V2 ] (y) = 0. (In this case, we show that G ∈ K2 ). Let X = {v ∈ V1 : deg[V1 ] (v) = 0}, Y = {v ∈ V2 : deg[V2 ] (v) = 0} and H = [X ∪ Y ∪ {v1 } ∪ {v2 }]. Then H is a 2K2 -free bipartite graph (else, P5 v G) with an edge v1 v2 which dominates G, and [(V1 \ X ) ∪ (V2 \ Y )] is a disjoint union of complete subgraphs of G. Therefore, G ∈ K2 . 3.2. Characterization of {P5 , K4 − e}-free graphs that contain an induced C5 In this section, we derive a characterization of {P5 , K4 − e}-free graphs that contain an induced C5 using a result proved by Arbib and Mosca [1]. The eleven graphs shown in Fig. 1 are {P5 , K4 − e}-free and contain an induced C5 (shown in bold lines). Let Dk denote the set of vertices at distance k from this C5 .
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Fig. 1. Basic graphs.
Theorem B (Theorem 1.6 of [1]). Any connected {P5 , K4 − e}-free graph that properly contains an induced C5 can be obtained from one of the graphs shown in Fig. 1 by expanding some v ∈ D1 ∪ C5 by independent sets and/or expanding some v ∈ D2 by a P3 -free graph. Note that in Figure 3 of [1], the edge a0 b0 is missing in G4 , and G6 is isomorphic to G7 . So, for convenience, we have renumbered the figures. More importantly note that every graph obtained as described in Theorem B is not a {P5 , K4 − e}free graph. For example, by expanding the vertex p of G6 (see Fig. 1) by an independent set of size ≥ 2, we get a graph which contains an induced K4 − e. In this respect, the following easy observation is useful to obtain the converse of Theorem B. Observation 1. Let G0 be a connected {P5 , K4 − e}-free graph on vertices v1 , v2 , . . . , vn that contains an induced C5 and let G = G0 (H1 , H2 , . . . , Hn ) be an expansion of G0 . Then G is a {P5 , K4 − e}-free graph that contains an induced C5 if and only if the expansion is obtained as follows. For any i, 1 ≤ i ≤ n:
• If vi has exactly one neighbor in G0 , then Hi is P3 -free. • If vi has two nonadjacent neighbors and has no two adjacent neighbors in G0 , then Hi is an independent set. • If vi has two nonadjacent neighbors and two adjacent neighbors in G0 , then Hi is K1 . So, combining Theorem B and Observation 1, we deduce the following result. Theorem 2. Let G be a connected graph. Then G is {P5 , K4 − e}-free and contains an induced C5 if and only if G is obtained from one of the basic graphs shown in Fig. 1 by expanding (i) each vertex shown under a circle by an independent set, and (ii) the vertex shown under a square by a P3 -free graph (see Fig. 1). We thank one of the referees who indicated Observation 1 and outlined its use in the proof of Theorem 2. This considerably shortened our original proof. 4. First-Fit coloring of graphs The First-Fit coloring algorithm (FFC) receives the vertices of a graph G in some order (v1 , v2 , . . . ). At any instant i, it performs two operations: (i) Receives vi and all the edges that vi is incident with v1 , v2 , . . . or vi−1 (but receives no information on which vertex succeeds vi ). (ii) Colors vi with the first available color (∈ {1, 2, 3, . . .}) that does not appear on any vertex adjacent with vi (and subsequently does not change this color). The number of colors used by the FFC depends on the order of the vertices it receives. The parameter χFF (G) denotes the maximum number of colors used by the FFC among all the orderings of the vertices of G. An on-line coloring algorithm (OLC) performs exactly as above except that while coloring vi in Step (ii), it need not select the first available color. Hence, FFC is an OLC, but an OLC need not be FFC. Given an ordering of the vertices of G, an OLC may use more than, less than or same number of colors as that of FFC. For example, the vertices of C5 (K2 , K2 , K2 , K2 , K2 ) can be ordered so that FFC performs poorer than OLC. For such comparisons, and for a survey on OLC and FFC; see [10,13]. Clearly,
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• The FFC of G induces a partition V (G) = (V1 , V2 , . . . , Vk ) such that (i) [Vi ] is independent, for all i (1 ≤ i ≤ k), and (ii) every vertex in Vj is adjacent to some vertex in Vi , for every i, j where 1 ≤ i < j ≤ k. • χFF (G) ≤ 1(G) + 1. • χFF (H ) ≤ χFF (G), for every H v G. If x ∈ V (G), let cFF (x) denote the color of x in a FFC of G. For S ⊆ V (G), define CFF (S ) := {cFF (v) : v is a vertex in [S ]}. We use often the following crucial observation which is implicit in the proof of Theorem 2.5 of [7]. Lemma 1 ([7]). Let S be any vertex subset of G. If [P ] and [Q ] are any two components of G − S, then CFF (S ∪ P ) ⊆ CFF (S ∪ Q ) or CFF (S ∪ Q ) ⊆ CFF (S ∪ P ). So, there exists a component D of G − S such that CFF (V (G)) ⊆ CFF (S ∪ D). An easy consequence of Lemma 1 by setting S = ∅ is the following. Lemma 2. For any graph G, χFF (G) = max{χFF (D) : D is a component of G}.
Next, we prove a lemma which enables us to relate χFF (G) and χFF (IG). Lemma 3. If u and v are two nonadjacent vertices in a graph G such that N (u) = N (v), then (i) cFF (u) = cFF (v), and (ii) χFF (G) = χFF (G − u). Proof. (i) Let χFF (G) = k and let (V1 , V2 , . . . , Vk ) be the partition of V (G) induced by the FFC. The assertion (i) is proved if we show that u, v ∈ Vi , for some i. Assume that u ∈ Vi and v ∈ Vj , where i < j. Since uv 6∈ E (G), FFC ensures that there exists a vertex x ∈ Vi such that xv ∈ E (G). Since N (u) = N (v), xu ∈ E (G), which contradicts that Vi is an independent set. (ii) Follows by (i). We also use the following theorem proved by Gyárfás and Lehel [11]. Theorem C ([11]). If G is a {P5 , K3 }-free graph, then χFF (G) ≤ 3.
4.1. First-Fit coloring of {P5 , K4 − e}-free graphs Theorem 3. If G is a {P5 , K4 − e}-free graph, then χFF (G) ≤ 2ω(G) − 1. Proof. Consider a FFC of G which uses χFF (G) colors and let (V1 , V2 , V3 , . . . , VχFF (G) ) be the consequent partition of V (G); so χFF (G) = |CFF (V (G))|. In view of Lemma 2, it is enough if we prove the theorem for connected graphs with clique number at least 2. If ω(G) = 2, then G is a {P5 , K3 }-free graph, and the result follows by Theorem C. In the following, all our graphs have ω(G) ≥ 3. (a) Assume that G is C5 -free. By Theorem 1, G ∈ K1 ∪ K2 ∪ MC. We follow the notation which we used in the definition of K1 , K2 and MC. (i) If G ∈ K1 , then by Lemma 1, there exists a complete graph Q in G − V (D) such that CFF (V (G)) ⊆ CFF (V (D) ∪ V (Q )). Since |CFF (V (D) ∪ V (Q ))| ≤ |V (D) ∪ V (Q )| ≤ 2ω(G) − 1, we have χFF (G) ≤ 2ω(G) − 1. (ii) If G ∈ K2 , then by Lemma 1, there exists a component P in G − {v1 , v2 } such that CFF (V (G)) ⊆ CFF (V (P ) ∪ {v1 , v2 }), where P is either a complete graph or a 2K2 -free bipartite graph. If P is a complete graph, then |CFF (V (G))| ≤ |CFF (V (P ) ∪ {v1 , v2 })| ≤ (ω(G) − 1) + 2 = ω(G) + 1 ≤ 2ω(G) − 1. If P is a 2K2 -free bipartite graph, we assume that |CFF (V (P ) ∪ {v1 , v2 })| ≥ 2ω(G), and arrive at a contradiction. Since ω(G) ≥ 3 and CFF (V (G)) ⊆ CFF (V (P ) ∪ {v1 , v2 }), there are at least six colors in CFF (V (P ) ∪ {v1 , v2 }). Let i, j, k and l be the colors in CFF (V (P )) \ CFF ({v1 , v2 }). Therefore, cFF (v1 ) 6= i, j, k, l and cFF (v2 ) 6= i, j, k, l. Let Ut = Vt ∩ V (P ), for t ∈ {i, j, k, l}. W. l. o. g., assume that i < j < k < l. Let x ∈ Uj . Then since j > i, by the FFC principle, there exist y ∈ Ui such that xy ∈ E (P ). Claim: If a ∈ Uk or Ul , then ax ∈ E (P ). On the contrary, assume that ax 6∈ E (P ). Then, there exists x0 ∈ Uj such that ax0 ∈ E (P ). Now, (1) if ay ∈ E (P ), then 0 x y 6∈ E (P ) (else, K3 v P). So, there exists y0 ∈ Ui such that x0 y0 ∈ E (P ). Then y0 a 6∈ E (P ) (else, K3 v P). Since P is 2K2 free, y0 x ∈ E (P ). But, then [{a, y, x, y0 , x0 }] ∼ = C5 v P, a contradiction. Next, (2) if ay 6∈ E (P ), then there exist y0 ∈ Ui and x0 ∈ Uj such that ay0 , ax0 ∈ E (P ). Since P is a 2K2 -free bipartite graph, xy0 , x0 y ∈ E (P ) and x0 y0 6∈ E (P ). But, then [{a, y0 , x, y, x0 }] ∼ = C5 v P, a contradiction. So, ax ∈ E (P ). Hence the claim holds. Let z ∈ Uk and t ∈ Ul . Then by above claim, zx, tx ∈ E (P ). Then tz 6∈ E (P ) (else, K3 v P). So, there exists z 0 ∈ Uk such that tz 0 ∈ E (P ). Then z 0 x 6∈ E (P ) (else, [{z 0 , x, t }] ∼ = K3 v P). So, there exists x0 ∈ Uj such that z 0 x0 ∈ E (P ). Since P is 2K2 -free, x0 z ∈ E (P ) and since it is bipartite x0 t 6∈ E (P ). But, then [{t , x, z , x0 , z 0 }] ∼ = C5 v P, a contradiction. Hence, |CFF (V (P ) ∪ {v1 , v2 })| ≤ 2ω(G) − 1, and so χFF (G) = |CFF (V (G))| ≤ |CFF (V (P ) ∪ {v1 , v2 })| ≤ 2ω(G) − 1. (iii) If G ∈ MC, then it is easy to see that deg (v) ≤ ω(G), for all v ∈ V (G). Hence, χFF (G) ≤ 1(G)+ 1 ≤ ω(G)+ 1 ≤ 2ω(G)− 1. (b) Assume that G contains an induced C5 . (In this case, we use Theorem 2.)
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Fig. 2. The vertex indicated in square is expanded by a P3 -free graph.
By Theorem 2, G is obtained from one of eleven basic graphs H as described in Theorem 2. If H 6= G10 , then by repeatedly applying Lemma 3(ii) to G, we have χFF (G) = χFF (H ). It can be easily verified that χFF (H ) ≤ 2ω(H ) − 1. Also, ω(G) = ω(H ), since the independent expansion does not change the clique number. Hence, χFF (G) ≤ 2ω(G) − 1. If H = G10 , then by repeatedly applying Lemma 3(ii) to G, we have χFF (G) = χFF (G∗ ), where G∗ is the graph shown in Fig. 2. Also, ω(G) = ω(G∗ ) as above. By Lemma 1, there exists a component D of G∗ − S, where S = {x, y, z }, such that CFF (V (G∗ )) ⊆ CFF (S ∪ V (D)). Since D is complete and |CFF (S ∪ V (D))| ≤ |S ∪ V (D)| ≤ 3 + (ω(G∗ ) − 1) ≤ 2ω(G∗ ) − 1, we have χFF (G) = χFF (G∗ ) ≤ 2ω(G∗ ) − 1 = 2ω(G) − 1. Remark. The bound given in Theorem 3 cannot be improved. To justify this claim, we construct a {P5 , K4 − e}-free graph G with ω(G) = t and present an ordering of its vertices for which the First-Fit coloring uses 2t − 1 colors.
• We first present ‘t’ vertex disjoint copies of Kt −1 , say Kt1−1 , Kt2−1 , . . . , Ktt−1 . The FFC assigns the colors 1, 2, 3, . . . , t − 1 to the vertices of Kti−1 (1 ≤ i ≤ t). • Next, we present a complete subgraph K ∼ = [{v1 , v2 , v3 , . . . , vt }] such that for each i (1 ≤ i ≤ t), (1) V (K ) ∩ V (Kti−1 ) = ∅, j i (2) [{vi }∪V (Kt −1 )] is a complete subgraph and (3) [{vi }, V (Kt −1 )] = ∅, for all j 6= i. Now, v1 , v2 , v3 , . . . , vt are respectively colored by t , t + 1, t + 2, . . . , 2t − 1 by the FFC. The graph G so constructed belongs to K1 , and hence it is {P5 , C5 , K4 − e}-free and ω(G) = t. 4.2. First-Fit coloring of {P5 , paw}-free graphs A characterization of paw -free graphs by Olariu [17] is the following. Theorem D ([17]). Let G be a connected graph. Then G is paw -free if and only if G is triangle-free or complete multipartite.
Theorem 4. If G is a {P5 , paw}-free graph, then χFF (G) ≤ ω(G) + 1. Proof. In view of Lemma 2, it is enough if we prove the theorem for connected graphs with clique number at least 2. If G is triangle-free, then the result follows by Theorem C. If G is a complete multipartite graph, then it is an independent expansion of a complete graph Q . So, by repeatedly applying Lemma 3(ii) to G, we have χFF (G) = χFF (Q ) = |Q | = ω(G). Given any ordering of the vertices of the {P5 , K3 }-free graph C5 (Kmc , Kmc , Kmc , Kmc , Kmc ), m ≥ 1, FFC uses three colors. Hence the upper bound given in Theorem 4 cannot be improved. Acknowledgements The authors gratefully acknowledge the referees for their suggestions and comments which improved the presentation of the paper. The second author acknowledges CSIR, India for the financial support to carry out this research work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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