Note on edge-colored graphs and digraphs without properly colored cycles

Note on edge-colored graphs and digraphs without prop erly colored cycles Gregor y Gutin Department of C omputer Science Ro y al Holl o w a y , Universit y of London Egham , Surrey , TW20 0EX, UK Gutin@cs.r h ul.ac. uk Abstract W e study the follo wing tw o functions: d ( n, c ) and ~ d ( n, c ); d ( n, c ) ( ~ d ( n, c )) is the minimum n um b er k su c h that ev ery c -edge-colored u ndirected (directed) graph of order n an d m inim um mono c hromatic d egree (out-degree) at least k has a prop erly colored cycle. Ab ouelaoualim et al. (2007) stated a conjecture whic h implies that d ( n, c ) = 1 . Using a recursiv e construction of c -edge-c olored graphs with m inim um mono chromatic degree p and without prop erly colored cycles, we sho w that d ( n, c ) ≥ 1 c (log c n − log c log c n ) and, th us, the conjecture do es not hold. In particular, this inequalit y significan tly impro v es a lo w er b ound on ~ d ( n, 2) obtained by Gutin, Sudako v and Y eo in 1998. Keywor ds: e dge-c olor e d gr aphs, pr op erly c olor e d cycles. 1 In tro du ction All directed and undirected g raphs considered in this pap er are simple, i.e., hav e no lo ops or parallel edges. W e consider only directed cycles in dig raphs; the term cycle (in a digraph) will alwa ys mean a directed cycle. Let G = ( V , E ) b e a directed or undirected gra ph, and let χ : E → { 1 , 2 , . . . , c } b e a fixed ( no t necessarily prop er) edge-coloring o f G with c colors, c ≥ 2. With giv en χ , G is called a c -e dge-c olo r e d (or , e dge-c olor e d ) graph. A subgraph H of G is called pr op erly c olor e d if χ defines a prop er edge-coloring of H , i.e., no v ertex 1 of H is inciden t to a pair of edges of the same color. F or a vertex of a c -edge- colored graph G , d i ( x ) denotes the n um b er of edges of color i inciden t with x . Let δ mon ( G ) = min { d i ( x ) : x ∈ V ( G ) , i ∈ { 1 , 2 , . . . , c }} . If G is directed, d + i ( x ) denotes the nu m b er of edges of color i in whic h x is tail. Let δ + mon ( G ) = min { d + i ( x ) : x ∈ V ( G ) , i ∈ { 1 , 2 , . . . , c }} . The authors of [2] stated the follow ing: Conjecture 1.1 L et G b e a c -e dge-c ol o r e d undir e cte d gr aph o f or der n with δ mon ( G ) = d ≥ 1 . Then G has a pr op erly c olor e d cycle of l e n gth at l e ast min { n, cd } . Mor e over, if c > 2 , then G has a p r o p erly c olor e d cycle of length at le ast min { n, cd + 1 } . In the next section, using a recursiv e construction of c -edge-colored gra phs with minim um mono c hro matic degree d and without prop erly colored cycles, w e sho w that t his conjecture do es no t hold. M oreo v er, for ev ery d ≥ 1 there exists an edge- colored graph G with δ mon ( G ) ≥ d and with no prop erly colored cycle. W e will study the following t w o functions: d ( n, c ) a nd ~ d ( n, c ); d ( n, c ) ( ~ d ( n, c )) is the minim um n um b er k suc h tha t ev ery c -edge-colored graph (digra ph) of order n and minimum mono chromatic degree (out-degree) at least k has a prop erly colored cycle. Gutin, Sudak o v and Y eo [5] prov ed the following b o unds for ~ d ( n, 2) 1 4 log 2 n + 1 8 log 2 log 2 n + Θ(1) ≤ ~ d ( n, 2) ≤ log 2 n − 1 3 log 2 log 2 n + Θ(1) (1) Using our construction, w e prov e that ~ d ( n, 2) ≥ 1 2 (log 2 n − log 2 log 2 n ). This impro v es the low er bound in (1). (The low er b ound in (1) w a s o btained using significan tly mor e elab orat e argumen ts.) This b ound on ~ d ( n, 2) fo llows from low er and upp er b ounds on d ( n, c ) and ~ d ( n, c ) obtained for eac h v alue of c. The b ounds imply that d ( n, c ) = Θ(lo g 2 n ) a nd ~ d ( n, c ) = Θ(log 2 n ) fo r each fixed c ≥ 2 . Prop erly colored cycles hav e b een studied in sev eral pap ers, for a surv ey , see Chapter 11 in [3]. Prop erly colored cycles in 2 -edge-colored undirected graphs gen- eralize cycles in digraphs and are of interest in genetics [3]. M ore recen t pap ers on prop orly colored cycles include [1, 2, 4]. Inte restingly , the pro blem to c hec k whether an edge-colored undirected graph has a prop erly colored cycle is p olynomial time solv able (w e can ev en find a shortest prop erly colored cycle is p olynomial time [1]), but the same problem for edge-color ed dig raphs is NP-complete [5]. 2 2 Results Theorem 2.1 F o r e ach d ≥ 1 ther e is an e dge-c ol o r e d gr aph G with δ mon ( G ) = d and w ith no pr op erly c olor e d cycle. Pro of: Let ( p 1 , p 2 , . . . , p c ) b e a v ector with nonnegativ e integral co ordinates p i . F or an arbitrary ( p 1 , p 2 , . . . , p c ), G ( p 1 , p 2 , . . . , p c ) is recursiv ely defined as follows : tak e a new vertex x a nd graphs H 1 = G ( p 1 − 1 , p 2 , p 3 , . . . , p c − 1 , p c ) if p 1 > 0, H 2 = G ( p 1 , p 2 − 1 , p 3 , . . . , p c − 1 , p c ) if p 2 > 0, . . . , H c = G ( p 1 , p 2 , p 3 , . . . , p c − 1 , p c − 1 ) if p c > 0 and add an edge of color i b et w een x and and every v ertex of H i for eac h i for whic h p i > 0 . In particular, G (0 , 0 , . . . , 0) = K 1 . It is easy to see, b y induction on p 1 + p 2 + · · · + p c , that G = G ( p 1 , p 2 , . . . , p c ) has no prop erly colored cycle a nd δ mon ( G ) = min { p i : i = 1 , 2 , . . . , c } . ✷ In fact, for each d ≥ 1 there are infinitely man y edge-colored g r a phs G with δ mon ( G ) = d and with no prop erly colored cycle. Inde ed, in the construction of G ( p 1 , p 2 , . . . , p c ) ab ov e we ma y assume that G (0 , 0 , . . . , 0) is an edgeless graph of arbitrary order. Lemma 2.2 L et n ( p 1 , p 2 , . . . , p c ) b e the o r der of G ( p 1 , p 2 , . . . , p c ) and let n c ( p ) = n ( p 1 , . . . , p c ) for p = p 1 = · · · = p c . Then n ( p 1 , . . . , p c ) ≤ s 2 s , wh er e s = p 1 + p 2 + . . . + p c , pr ovide d s > 0 an d p ≥ 1 c (log c n c ( p ) − log c log c n c ( p )) . Pro of: W e first pro v e n ( p 1 , . . . , p c ) ≤ s 2 s b y induction on s ≥ 1 . The inequalit y clearly holds for s = 1. By induction hypothesis, for s ≥ 2, w e hav e n ( p 1 , . . . , p c ) ≤ 1 + X { n ( p 1 , . . . , p i − 1 , p i − 1 , p i +1 , . . . , p c ) : p i > 0 , i = 1 , 2 , . . . , c } ≤ 1 + c ( s − 1) c s − 1 ≤ sc s Th us, n c ( p ) ≤ cp · c cp . Observ e that n c ( p ) > ac a pro vided a = log c n c ( p ) − log c log c n c ( p ) and, th us, cp ≥ log c n c ( p ) − log c log c n c ( p ) . ✷ Corollary 2.3 We have ~ d ( n, c ) ≥ d ( n, c ) ≥ 1 c (log c n − log c log c n ) . Pro of: Let H b e a c -edge-colored undirected g r aph and H ∗ b e a digraph obtained from H b y replacing ev ery edge e = xy with arcs xy and y x b oth of color χ ( e ) . 3 Clearly , H has a prop erly colored cycle if and only if H ∗ has a prop erly colored cycle. Thus , ~ d ( n, c ) ≥ d ( n, c ). The inequalit y d ( n, c ) ≥ 1 c (log c n − log c log c n ) follows from Lemma 2.2 and the fact that graphs G ( p, p, . . . , p ) ha ve no prop erly colored cycles. ✷ W e see tha t ~ d ( n, 2) ≥ 1 2 (log 2 n − log 2 log 2 n ). This is an impro v ement ov er the lo w er b ound on ~ d ( n, 2) in (1). Using the upp er b ound in (1), w e will obtain an upp er b ound on ~ d ( n, c ) and, thus , d ( n, c ) . Prop osition 2.4 We have ~ d ( n, c ) ≤ 1 ⌊ c/ 2 ⌋ (log 2 n − 1 3 log 2 log 2 n + Θ(1)) . Pro of: Let D b e a c -edge-colored digraph of order n with δ mon ( D ) ≥ 1 ⌊ c/ 2 ⌋ (log 2 n − 1 3 log 2 log 2 n + Θ(1)) . Let D ′ b e the 2-edge-colored digraph obtained from D b y assigning color 1 to all edges of D o f color 1 , 2 , . . . , ⌊ c/ 2 ⌋ and color 2 to all edges of D of color ⌊ c/ 2 ⌋ + 1 , ⌊ c/ 2 ⌋ + 2 , . . . , c. It remains to observ e that δ mon ( D ′ ) ≥ log 2 n − 1 3 log 2 log 2 n + Θ(1 ) and ev ery prop erty colored cycle in D ′ is a pr o p ert y colored cycle in D . ✷ Corollary 2.5 F or every fix e d c ≥ 2 , we have d ( n, c ) = Θ(log 2 n ) a nd ~ d ( n, c ) = Θ(log 2 n ) . 3 Op e n Probl ems W e b eliev e that there a re functions s ( c ) , r ( c ) dep enden t only on c suc h tha t d ( n, c ) = s ( c ) log 2 n (1 + o (1)) and ~ d ( n, c ) = r ( c ) log 2 n (1 + o (1)) . In particular, it w ould b e in t eresting to determine s ( 2 ) a nd r (2) . References [1] A. Ab ouelaoualim, K.Ch. Das, L. F aria, Y. Manoussakis, C.A. Martinhon and R. Saad, P aths and trails in edge-color ed g raphs. Submitted, 2007 . [2] A. Ab ouelaoualim, K .Ch. Das, W. F ernandez de la V ega, M. Karpinski, Y. Manoussakis, C.A. Martinhon and R . Saad, Cycles a nd paths in edge-colored graphs with giv en degrees. Submitted, 2007. 4 [3] J. Bang-Jensen and G . Gutin, Digr aphs: The ory, A lgorithms and Applic ations , Springer-V erlag, London, 2000. [4] H. Fleisc hner and S. Szeider, On Edge-Colored Graphs Co ve red by Prop erly Colored Cycles. Graphs and Com binatorics 21 (2 0 05), 301–306 . [5] G. Gutin, B. Sudako v a nd A. Y eo, Note on a lt ernat ing directed cycles. Discrete Math. 191 (1998) , 101-1 07. 5