Edge-coloring series-parallel multigraphs

Edge-Coloring Series-P arallel Multigraphs Cristina G. F ernandes ∗ Departamento de Ciˆ encia da Comput a¸ c˜ ao Instituto de Matem´ atica e Estat ´ ıstica Universidade de S˜ ao Pa u lo - Brazil E-mail: cris@ime.usp.br and Robin Thomas † School of Mathematics Georgia Institut e of T ec h nology Atlan ta, GA 30332-0160, USA E-mail: thomas@math.gatec h .edu Septem b er 14, 2018 Abstract W e give a simpler proof of Seymour’s Theor em on edge-colo ring series-pa rallel multigraphs and der ive a linear-time algor ithm to chec k whether a given series- parallel m ultig r aph ca n b e colored with a given num b er of colo rs. 1 In tro du ction All gr aphs in this pap er are finite, ma y ha ve parallel edges, but no loop s . Let k ≥ 0 b e an in teger. A graph G is k - e dge- c olor able if th ere exists a map κ : E ( G ) → { 1 , . . . , k } , calle d a k -e dge-c oloring , suc h that κ ( e ) 6 = κ ( f ) for an y t wo distinct edges e, f of G that share at least one end. The chr omatic index χ ′ ( G ) is the minimum k ≥ 0 such that G is k -edge-colorable. Clearly χ ′ ( G ) ≥ ∆( G ), where ∆( G ) is the maxim um degree of G , but there is another lo w er b oun d . Let Γ( G ) = m ax  2 | E ( G [ U ]) | | U | − 1 : U ⊆ V ( G ) , | U | ≥ 3 and | U | is o dd  . If U is as ab o v e, then every matc h ing in G [ U ], the subgraph indu ced by U , has size at most ⌊ 1 2 | U |⌋ . Consequent ly , χ ′ ( G ) ≥ Γ( G ). If G is the Pete r sen graph, or th e Petersen graph with one vertex deleted, then χ ′ ( G ) > max { ∆( G ) , ⌈ Γ( G ) ⌉} . Ho w ever, S eymour conjectures that equalit y holds for planar graphs: Conjecture 1.1 If G is a planar gr aph, then χ ′ ( G ) = m ax { ∆( G ) , ⌈ Γ( G ) ⌉ } . Conjecture 1. 1 most lik ely do es not ha ve an easy p r o of, b ecause it implies the F our-Color Theorem. Marcotte [5] prov ed that this conjecture holds for graphs w hic h do not contai n K 3 , 3 and do not con tain K 5 \ e as a minor (where K 5 \ e is the graph obtained from K 5 b y remo ving one of its edges). This resu lt extended a pr evious result by S eymour [6], w ho prov ed that his conjecture holds for series-parallel graph s (a graph is series-p ar al lel if it has no subgraph isomorph ic to a su b division of K 4 ): Theorem 1.2 If G i s a series-p ar al lel gr aph, and k is an inte ger with k ≥ max { ∆( G ) , Γ( G ) } then G i s k - e dge-c olor able. ∗ Researc h sup p orted in part by CNPq Proc. N o. 301174/97 - 0, F APESP Pro c. No. 98/143 29 and 96/04505-2 and PRONEX/CNPq 664107/ 1997-4 (Brazil). † Researc h supp orted in part by NSA u n der Grant No. MDA904-98-1-0517 and b y NSF u nder Gran t No. DMS- 9970514 . 1 It shou ld b e noted th at Theorem 1.2 is fairly easy for simple graphs; the difficult y lies in the presence of parallel edges. Seymour’s pro of is elegan t and in teresting, bu t the indu ction step requ ires the verificat ion of a large num b er of inequalities. W e giv e a simp ler pro of, based on a stru ctur al lemma ab out series-parallel graphs, whic h in turn is an easy consequence of the well- k n o wn fact that ev ery simple series-parallel graph has a vertex of degree at most tw o. Our w ork was motiv ated b y the list edge-coloring conjecture of [1] (see also [3, Problem 12.20]): Conjecture 1.3 E very gr aph is χ ′ ( G ) -e dge-cho osable. A t present t h ere seems to b e no cred ible approac h for pro ving the conjecture in full generalit y . W e were trying to gain some insight by stu d ying it for series-parallel graphs. The conjecture has b een v erified for simple series-parallel graphs in [4 ], b u t it is op en for series-parallel graph s with parallel edges. Our efforts only resulted in a simpler pro of of T h eorem 1.2 and in a linear-time algorithm f or chec king w hether or n ot a s eries-parallel graph can b e colored with a giv en n umber of colors. Our algorithm substantia lly simplifi es an earlier algorithm of Zh ou, Su zuki and Nishizeki [7]. 2 Three lemmas F or our pro of of Theorem 1.2 w e need three lemmas. The first tw o are easy , and the third app eared in [4]. Let G b e a graph, and let u, v b e adj acen t vertice s of G . W e us e uv to denote the unique edge with ends u and v in the underlying simple g r aph of G . If G has m e d ges with ends u and v , then we sa y that uv has multiplicity m . If u and v are not adjacen t, then w e sa y that uv has m ultiplicit y zero. Let G b e a graph, le t κ b e a k -edge-coloring o f a sub grap h H of G , let u ∈ V ( G ), and let i ∈ { 1 , 2 , . . . , k } . W e sa y that u se es i and that i is se en b y u if κ ( f ) = i for some edge f of H in cident with u . Lemma 2.1 L et G b e a gr aph, let u 0 ∈ V ( G ) , let u 1 , u 2 b e distinct neighb ors of u 0 , let H b e the g r aph obtaine d fr om G by deleting al l e dges with one end u 0 and the other e nd u 1 or u 2 , and let κ b e a k -e dge-c oloring of H . F or i = 1 , 2 let m i b e the multiplicity of u 0 u i in G , and for i = 0 , 1 , 2 let S i b e the set of c olors se en by u i . If m 1 + | S 0 ∪ S 1 | ≤ k , m 2 + | S 0 ∪ S 2 | ≤ k and m 1 + m 2 + | S 0 ∪ ( S 1 ∩ S 2 ) | ≤ k , then κ c an b e extende d to a k -e dge-c oloring of G . Pro of. Since m 1 + | S 0 ∪ S 1 | ≤ k , the edges with ends u 0 and u 1 can be colored using colors not in S 0 ∪ S 1 . W e do that, using as many colors in S 2 as p ossible. If the u 0 u 1 edges can be co lored using colors in S 2 only , then there are at least k − | S 0 ∪ S 2 | ≥ m 2 colors left to color the edges w ith ends u 0 and u 2 , and so κ can b e exte n ded to a k -edge-coloring of G , as desired. Otherwise, the u 0 u 1 edges of G will b e colored usin g | S 2 \ ( S 0 ∪ S 1 ) | colors fr om S 2 , and m 1 − | S 2 \ ( S 0 ∪ S 1 ) | other colors. Thus the n u m b er of colors a v ailable to color the u 0 u 2 edges of G is at least k − | S 0 ∪ S 2 |− ( m 1 − | S 2 \ ( S 0 ∪ S 1 ) | ) = k − m 1 − | S 0 ∪ ( S 1 ∩ S 2 ) | ≥ m 2 , a n d s o the coloring can b e c omp leted to a k -edge-coloring of G , a s desired. Lemma 2.2 L et k b e an inte ger, and let G b e a gr aph with ∆( G ) ≤ k . Then Γ( G ) ≤ k if and only if 2 | E ( G [ U ]) | ≤ k ( | U | − 1) for every set U ⊆ V ( G ) such that | U | is o dd and at le ast thr e e, and the underlying g r aph of G [ U ] has no v e rtic es of de gr e e at most one. Pro of. T he “only if ” part is clear. T o p ro ve the “if ” part w e m u s t sho w that 2 | E ( G [ U ]) | ≤ k ( | U | − 1) for eve r y set U ⊆ V ( G ) such that | U | is o d d and at least th r ee. W e pro ceed b y ind uction on | U | . W e ma y assume t h at the un derlying graph of G [ U ] has a v ertex u of degree at most one, for o th erwise the conclusion follo w s from the hyp othesis. If u has d egree one in the underlyin g graph of G [ U ], then let v b e its unique neigh b or; otherwise let v ∈ U \{ u } b e arbitrary . Let U ′ = U \{ u, v } . Then 2 | E ( G [ U ]) | ≤ 2∆( G ) + 2 | E ( G [ U ′ ]) | ≤ 2 k + k ( | U ′ | − 1) ≤ k ( | U | − 1) by the indu ction h yp othesis if 2 | U | > 3 and triviall y otherwise, as desired. The third lemma app eared in [4]. F or the sak e of completeness we include its short pro of. Lemma 2.3 Ev ery non-nul l simple serie s- p ar al lel gr aph G has one of the fol lowing: (a) a vertex of de gr e e at most one, (b) two distinct vertic es of de gr e e two with the same neighb ors, (c) two distinct ve rtic es u, v and two not ne c essarily distinct vertic es w , z ∈ V ( G ) \{ u, v } such that the neighb ors of v ar e u and w , and e very neighb or of u is e qu al to v , w , or z , or (d) five distinct vertic es v 1 , v 2 , u 1 , u 2 , w such that the neighb ors of w ar e u 1 , u 2 , v 1 , v 2 , and for i = 1 , 2 the neig hb ors of v i ar e w and u i . Pro of. W e pr o ceed by induction on the n umb er of v ertices. Let G b e a non-n u ll simple series- parallel graph , an d assume that the r esult holds for all graphs on fewer v ertices. W e may assume that G d o es n ot satisfy (a), (b), or (c). Th us G has n o t wo adjacen t vertice s of d egree tw o. By suppr essing all vertice s of degree t wo (that is, con tracting one of the inciden t edges) w e obtain a series-parallel graph without v ertices of d egree tw o or less. Therefore, b y a well -kn o wn prop ert y of series-parallel g r aphs [2], this graph is not simple. S ince G d o es n ot satisfy (b), t h is imp lies that G has a v ertex of degree tw o that b elongs to a cycle of length three. Let G ′ b e obtained fr om G b y deleting all v ertices of degree t wo that b elong to a cycle of length three. First notice that if G ′ has a v ertex of degree less than tw o, then the result holds for G (cases (a), (b), or case (c) w ith w = z ). Similarly , if G ′ has a v ertex of degree t wo that do es not hav e degree t wo in G , then th e result h olds (one of the cases (b)–(d) occur s ). T hus we ma y assume that G ′ has min im um degree at least t wo, and every ve r tex of degree t wo in G ′ has degree tw o in G . By indu ction, (b), (c), or (d) holds for G ′ , but it is ea s y to see that then one of (b), (c), or (d) holds for G . 3 Pro of of T heorem 1.2 W e pro ceed b y induction on | E ( G ) | , and , sub ject to that, by induction on | V ( G ) | . The theorem clearly holds for graphs with no edges, s o we assume th at G has at least one edge, and that the theorem holds for graphs with f ew er edges or the same n u m b er of edges b u t few er v ertices. Let S b e the und erlying simple graph of G . W e apply Lemma 2.3 to S , and distinguish the corr esp onding cases. If case (a) holds, let G ′ b e the graph obtained from G by remo ving a v ertex of degree at most one in S . The rest is straigh tforward: k ≥ max { ∆( G ′ ) , Γ( G ′ ) } and so, by in duction, ther e is a k -edge-colo r ing of G ′ . F r om this k -edge-colo r in g, it is easy to obtai n a k -edge-colo r ing for G . If case (b) holds, let u and v b e t w o d istinct vertice s of degree t w o in S with the same neigh- b ors. Let the common neigh b ors b e x and y . Let a, b, c, d b e the multipliciti es of ux, uy , v x , v y , resp ectiv ely . See Figure 1(a). F rom the symmetry w e ma y assum e that a ≥ d . Let G ′ b e obtained from G \ v by deleting d edges with ends u and x , and add in g d edges with end s u and y . See Figure 1(b). Th en clearly ∆( G ′ ) ≤ k , and it follo ws from Lemma 2.2 that Γ( G ′ ) ≤ k . By the induction h yp othesis the graph G ′ has a k -edge-col orin g κ ′ . Let A b e a set of colors of size d used b y a subset of the edges of G ′ with ends u and y , c h osen so that as few as p ossible of these colors are seen by x . By d eleting those edges we obtain a coloring of G \ v , where d edges with ends u and x are uncolored. Next we color those d uncolored edges, fi rst using colors in A not seen by x , and th en using arbitrary colors n ot seen by x or u . This can b e done: if at least one color in A is seen by x , then once w e exhaust colors of A not s een by x , the choic e of A implies that ev ery 3 color seen by u is seen b y x , and so the coloring can b e completed, b ecause x has degree at most k . Th is results in a k -edge-co lorin g of G \ v w ith the p rop erty that at least d of the colors seen b y x (namely the colors in A ) are not seen by y . Thus the n umb er of colors seen by b oth x and y is at most k − c − d ( v sees no colors), and clearly th e num b er of colors seen b y x is at most k − c and the n umb er of colors seen b y y is at most k − d . By Lemma 2.1 this colo r ing can b e extended to a k -edge-col orin g of G , as desired. W e no w assume a sp ecial case of (c) of Lemm a 2.3. Let u, v , w , z b e as in that lemma, with w = z . Then clea r ly ∆( G \ v ) ≤ k and Γ( G \ v ) ≤ k , and so G \ v has a k -edge-coloring. Th is k -edge- coloring can b e extended to a k -edge-colo r ing of G by fi rst coloring the edges with end s w and v (this can b e done b ecause the degree of w is at most k ), and then coloring the edges with ends u and v (there are enough colo r s for this b ecause | E ( G [ U ]) | ≤ k for U = { u, v , w } ). Finally w e assume that case (d) of Lemma 2.3 h olds and we will show that ou r analysis includes the remainder of case (c) as a sp ecial case. Let v 1 , v 2 , u 1 , u 2 and w b e as in th e statemen t of Lemma 2.3, and let a , b , c , d , e and f b e the m ultiplicities of u 1 v 1 , u 1 w , v 1 w , v 2 w , u 2 w and u 2 v 2 , resp ectiv ely , as in Figure 2(a). In order to includ e case (c) w e will not b e assum in g that a , b , c , d , e and f are nonzero; w e only assume th at c + d > 0. (This is wh y the primary indu ction is on | E ( G ) | .) If a + b + c + d + e + f ≤ k , then a k -edge-coloring of G \{ v 1 w, v 2 w } can b e extended to a k -edge-coloring of G , and so w e ma y assum e that k < a + b + c + d + e + f . Since w has degree at most k we h a v e b + c + d + e ≤ k , and b y considering th e sets U = { u 1 , v 1 , w } and U = { u 2 , v 2 , w } w e deduce that a + b + c ≤ k and d + e + f ≤ k . Let z 1 = max { 0 , a + b + c + e − k } , z 2 = max { 0 , b + d + e + f − k } and s = k − ( b + c + d + e ). Thus z 1 ≤ e , z 2 ≤ b , s ≥ 0 and a + f − z 1 − z 2 − s =        k − ( b + e ) if z 1 > 0 and z 2 > 0 a + c if z 1 = 0 and z 2 > 0 d + f if z 1 > 0 and z 2 = 0 a + f − s if z 1 = z 2 = 0. (1) W e claim that there exist n on n egativ e intege r s s 1 and s 2 suc h that s = s 1 + s 2 , s 1 ≤ a − z 1 and s 2 ≤ f − z 2 . T o pro ve this claim it suffices to c hec k that a − z 1 ≥ 0, f − z 2 ≥ 0 and a − z 1 + f − z 2 ≥ s . W e ha v e a − z 1 ≥ min { a, k − ( b + c + e ) } ≥ min { a, d } ≥ 0, and by symmetry f − z 2 ≥ 0. The third inequalit y follo ws from (1). T his pro v es the existence of s 1 and s 2 . Let G ′ b e obtained from G by remo vin g the vertices v 1 , v 2 , w , adding tw o new v ertices, x and y , and adding a − z 1 − s 1 edges with ends x and u 1 , f − z 2 − s 2 edges with ends x and u 2 , b − z 2 edges with ends y and u 1 , e − z 1 edges with end s y and u 2 , and z 1 + z 2 edges with ends u 1 and u 2 . See Figure 2(b). T h u s | E ( G ′ ) | < | E ( G ) | . It follo ws from (1) th at x has degree at most k . Since all other vertic es of G ′ clearly ha ve degree at most k , we see th at k ≥ ∆( G ′ ). W e claim that k ≥ Γ ( G ′ ). By Lemma 2.2 w e must P S f r a g r e p l a c e m e n t s (a) (b) G G ′ u u v x x y y a b c d a − d b + d Figure 1: C onfigurations referring to Case (b). 4 P S f r a g r e p l a c e m e n t s (a) (b) u 1 u 1 u 2 u 2 v 1 v 2 x y w a b c d e f a − z 1 − s 1 f − z 2 − s 2 z 1 + z 2 b − z 2 e − z 1 Figure 2: C onfigurations referring to Case (d). sho w that 2 | E ( G ′ [ X ′ ]) | ≤ k ( | X ′ | − 1) for every set X ′ ⊆ V ( G ′ ) su c h that | X ′ | is o d d, | X ′ | ≥ 3 and the under lyin g graph of G ′ [ X ′ ] has no v ertices of degree at most one. If | X ′ ∩ { u 1 , u 2 }| ≤ 1, then G [ X ′ ] = G ′ [ X ′ ], and the result follo w s. Th us we may assum e that u 1 , u 2 ∈ X ′ . W e need to distinguish sev eral cases. If x, y ∈ X ′ , then let X = X ′ \ { x, y } . W e hav e 2 | E ( G ′ [ X ′ ]) | = 2 | E ( G [ X ]) | + 2( a − z 1 − s 1 + f − z 2 − s 2 + z 1 + z 2 + b − z 2 + e − z 1 ) ≤ k ( | X ′ | − 1), using th e induction h yp othesis and the relations s 1 + s 2 = k − ( b + c + d + e ), z 1 ≥ a + b + c + e − k and z 2 ≥ b + d + e + f − k . If x ∈ X ′ and y 6∈ X ′ w e put X = X ′ \ { x } ∪ { w, v 1 , v 2 } , and if x 6∈ X ′ and y ∈ X ′ w e p u t X = X ′ \ { y } ∪ { w } . In either of these tw o cases the counting is straigh tforwa r d. Finally , w e assume that x, y 6∈ X ′ . If z 1 = z 2 = 0, then G [ X ′ ] = G ′ [ X ′ ], and so the conclusion holds . If z 1 > 0 and z 2 > 0, then let X = X ′ \ { u 1 , u 2 } . W e hav e 2 | E ( G ′ [ X ′ ]) | ≤ 2 | E ( G [ X ]) | + 2( k − ( a + b ) + k − ( e + f ) + z 1 + z 2 ) ≤ k ( | X | − 1) + 2( b + c + d + e ) ≤ k ( | X ′ | − 1), where the s econd inequalit y follo w s from the ind uction hyp othesis (or is trivial if | X | = 1) and the defin ition of z 1 and z 2 . Finally , fr om the symmetry b et w een z 1 and z 2 it suffices to consider t h e c ase z 1 = 0 and z 2 > 0. In that case w e put X = X ′ ∪ { w , v 2 } . Then 2 | E ( G ′ [ X ′ ]) | = 2 | E ( G [ X ]) | + 2( z 1 + z 2 − ( b + d + e + f )) ≤ k ( | X ′ | − 1), using the induction h y p othesis and the d efinition of z 1 and z 2 . This completes the pro of th at k ≥ Γ( G ′ ). By induction there exists a k -edge-coloring κ ′ of G ′ . Let Z 1 ∪ Z 2 b e the colors used on the z 1 + z 2 edges of E ( G ′ ) \ E ( G ) with ends u 1 and u 2 , so th at | Z 1 | = z 1 and | Z 2 | = z 2 . Let G ′′ b e the graph obtained from G by deleting all edges with one end w and the other en d v 1 or v 2 . W e first construct a suitable k -edge-colo r ing κ ′′ of G ′′ . T o do so w e start with the restriction of κ ′ to E ( G ′′ ) ∩ E ( G ′ ), and then use Z 1 and th e colors of the xu 1 edges of G ′ to color a s ubset of the u 1 v 1 edges of G , we us e Z 2 and the colors of the y u 1 edges of G ′ to co lor all of the w u 1 edges of G , an d symmetrically w e use Z 1 and th e colo r s of the u 2 y edges of G ′ to co lor all the w u 2 edges of G , and w e u s e Z 2 and all the colors of th e x u 2 edges of G ′ to color a sub set of the v 2 u 2 edges of G . W e color the s 1 uncolored u 1 v 1 edges and the s 2 uncolored u 2 v 2 edges arbitrarily . That can b e done, b ecause u i is the only neigh b or of v i in G ′′ . This completes th e definition of κ ′′ . No w the num b er of colors seen by v 1 or w is at most a − z 1 − s 1 + z 1 + z 2 + b − z 2 + e − z 1 + s 1 = a + b + e − z 1 ≤ k − c , and similarly the num b er of colors seen b y v 2 or w is at most k − d . The n umb er of colors seen b y w , or by b oth v 1 and v 2 is at most b − z 2 + e − z 1 + z 1 + z 2 + s ≤ k − ( c + d ). By Lemma 2.1 the k -edge-colo r ing κ ′′ can b e extended to a k -edge-col orin g of G , as desired. 4 A linear-time algorithm In this section w e p resen t a linear-time algorithm to d ecide whether χ ′ ( G ) ≤ k , wh ere the series- parallel graph G and the intege r k are part of th e inp u t instance. Th e idea of the algorithm is 5 v ery sim p le – we rep eatedly fi n d v ertices of the un derlying simple graph satisfying one of (a)–(d) of Lemma 2.3, constru ct the graph G ′ as in the pr o of of Theorem 1.2, apply th e algorithm recursivel y to G ′ to c hec k w hether χ ′ ( G ′ ) ≤ k , and fr om that kno wledge we d educe wh ether χ ′ ( G ) ≤ k . The construction of G ′ is straigh tforward, and the d ecision whether χ ′ ( G ) ≤ k is easy: su pp ose, for instance, th at w e fin d v ertices v 1 , v 2 , u 1 , u 2 , w as in Lemma 2.3(d), and let a, b, c, d, e , f b e as in the pro of of T heorem 1.2. If a + b + c + d + e + f ≥ k , then constru ct G ′ as in the pro of; w e ha ve χ ′ ( G ) ≤ k if and only if χ ′ ( G ′ ) ≤ k and a + b + c ≤ k and d + e + f ≤ k . If a + b + c + d + e + f ≤ k , then χ ′ ( G ) ≤ k if and only if χ ′ ( G \ w ) ≤ k . Th u s it remains to describ e ho w to find th e vertices as in Lemma 2.3. That can b e done b y a slight mo dification of a linear-ti m e recog n ition algorithm for series-parallel graphs. W e need a few definitions in order to describ e the algorithm. Let H b e a graph, and let λ b e a function assigning to eac h edge e ∈ E ( H ) a set λ ( e ) d isjoin t from V ( H ) in suc h a wa y th at λ ( e ) ∩ λ ( e ′ ) = ∅ for distinct edges e, e ′ ∈ E ( H ). Let H λ b e the graph obtained from H b y adding, for eac h edge e ∈ E ( H ) and eac h x ∈ λ ( e ), a vertex x of degree tw o, adjacen t to the t wo ends of e . Then H λ is un ique up to isomorphism, and so we can sp eak of the graph H λ . No w let µ : E ( H λ ) → Z + 0 b e a function, and let H µ λ b e the graph obtained from H λ b y replacing eac h edge e ∈ E ( H λ ) by µ ( e ) parallel edges with the same ends. In those circumstances w e say that ( H, λ, µ ) is an enc o ding , and that it is an enc o ding of H µ λ . F or a graph H and v ∈ V ( H ) w e let deg H ( v ) denote the n u m b er of edges in ciden t to v in H and v al H ( v ) denote the num b er of distinct neigh b ors of v in H . Thus v al H ( v ) ≤ d eg H ( v ) with equalit y if and o n ly if v is inciden t with no p arallel edges. W e sa y that a function C : V ( H ) → Z + 0 is a c ounter for a graph H if deg H ( v ) − v al H ( v ) ≤ C ( v ) for ev ery v ertex v ∈ V ( H ). W e sa y that a v ertex v ∈ V ( H ) is active if eit h er deg H ( v ) ≤ 2 or deg H ( v ) ≤ 3 C ( v ). The follo wing lemma guarantee s that if there are no act ive v er tices, then the graph is n ull. Lemma 4.1 L et H b e a non-nul l series-p ar al lel gr aph, and let C b e a c ounter for H . Then ther e exists an active vertex. Pro of. As noted in the p ro of of Lemma 2.3, the underlying simp le graph of H has a vertex of degree at most t w o. Th u s H has a v ertex v with v al H ( v ) ≤ 2. If deg H ( v ) > 3 C ( v ), th en deg( v ) − 2 ≤ d eg H ( v ) − v al H ( v ) ≤ C ( v ) < deg H ( v ) / 3 , whic h implies d eg H ( v ) ≤ 2. Thus v is activ e, as desired. 4.1 The algorithm The in put for the algorithm is a s eries-parallel graph G and a non-negativ e in teger k , where the graph G is presente d b y means of its underlying u ndirected graph and a function E ( G ) → Z + that describ es the m ultiplicit y of eac h edge. The algorithm starts by chec king whether deg G ( v ) ≤ k for all v ∈ V ( G ). If not, it outputs “no, χ ′ ( G ) 6≤ k ” and term in ates. Otherwise let H b e the underlyin g und irected graph of G , let λ ( e ) := ∅ for ev ery edge e ∈ E ( H ), let µ ( e ) b e the multiplici ty of e in G , and let C ( v ) := 0 for eve r y v ∈ V ( H ). Then ( H , λ, µ ) is an encodin g of G and C is a coun ter f or H . T he algorithm computes the list of all activ e vertice s of H . I t do es n ot matter ho w L is implemen ted as long as elemen ts can b e deleted and added in constan t time. After this, the algorithm is iterativ e. Eac h iteration starts with an enco d ing ( H , λ, µ ) of the current series-parallel graph G , a coun ter C for H and a list L w hic h includes all activ e v ertices of H . 6 Eac h iteration consists of the follo wing. If L = ∅ , then w e outp ut “y es, χ ′ ( G ) ≤ k ” and terminate, else w e let v b e a v ertex in L . If v 6∈ V ( H ) or v is not activ e, then w e remo ve v from L and mo ve to the n ext iteration. If v ∈ V ( H ) and v is activ e, then there are three p ossible cases. If deg H ( v ) > 2, then d eg H ( v ) ≤ 3 C ( v ), b ecause v is activ e. W e r earrange th e adj acency list of v , remo ving all but one edge from eac h class of p arallel edges inciden t with v , adjusting λ and µ so that ( H , λ, µ ) is stil l an encod ing of G . W e set C ( v ) := 0, include in L all v ertices whose degree decreased and mo ve to the next ite r ation. If d eg H ( v ) = v al H ( v ) = 2 and λ ( v x ) = λ ( v y ) = 0, where x and y are the tw o distinct neigh b ors of v , then w e remo v e v from H and add a n ew edge f = xy to H . W e set µ ( f ) := 0 , λ ( f ) := { v } , increase b oth C ( x ) a n d C ( y ) b y one, add x and y to L and mo v e to the n ext iteration. If deg H ( v ) ≤ 2 but the previous case do es not apply , then we hav e lo cated vertices of G satisfying one of (a) to (d) of L emm a 2.3. W e c h ec k if the lo cal conditions are satisfied or not (for example, in case (d), if a + b + c + d + e + f ≥ k , w e c heck whether a + b + c ≤ k and d + e + f ≤ k ); if they are not, w e outpu t “n o, χ ′ ( G ) 6≤ k ” and terminate. Otherwise, we m o dify the enco ding ( H , λ, µ ) to get an enco din g of the graph G ′ describ ed in the pro of of T h eorem 1.2. T his in volv es deleting v ertices from H and addin g edges to H . Every time an edge of H incident with a v ertex z ∈ V ( H ) is deleted or added w e increase C ( z ) by one and add z to L . W e mo v e to the next iterati on . The correctness of the algo r ithm f ollo w s from Lemma 4.1 and from the pro of of Theorem 1. 2 . T o analyze th e r unning-time, let n d enote the num b er of v er tices of the inpu t graph G . The initial steps of the algorithm can b e d on e in O ( n ) time. Eac h iteration takes time prop ortional to the decrease in the quan tity 2 K · | V ( H ) | + K · X e ∈ E ( H ) λ ( e ) + | L | + 4 · X v ∈ V ( H ) C ( v ) , where K is a sufficien tly large constan t. Thus the ru nning-time of the algo r ithm is O ( n ). References [1] B. Bollob´ as, A. J. Harris, List colorings of graphs , Graphs and Combinatorics 1 (1985), 115–1 27. [2] J. Duffin , T op ology of series-parallel net w orks , Journal o f Mathematical Analysis and Applica- tions 10 (1965 ) 303–318 . [3] T. R. Jensen, B. T oft, Graph Coloring Problems , Wiley , New Y ork, 1995. [4] M. Juv an, B. Mohar, and R. T h omas, List Ed ge-Colo r ings of S eries-P arallel Gr aphs , Electronic Journal of Com binatorics 6 (1999), no. 1, Researc h Pap er 42. [5] O. Marcotte, Op timal Ed ge-Colourings for a Class of Planar Multigraphs , Com b inatorica 21 (3) (200 1) 361–39 4. [6] P .D. Seymour, Colouring Series-P arallel Graphs , Com bin atorica 10 (4 ) (1990) 379–3 92. [7] X. Zh ou, H. Suzuki, and T. Nish izeki, A Linear-Time Algorithm for Edge-Col orin g Series- P arallel Multigraphs , J ournal of Algo r ithms 20 (1996), 174–20 1. This material is based up on w ork sup p orted by the National Science F oundation under Gran t No. DMS-9970514. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science F oundation. 7