Homomorphisms from sparse graphs with large girth

Transcription

1 Homomorphisms from sparse graphs wih large girh O.V. Borodin, S.-J. Kim, A.V. Kosochka, D.B. Wes revised April, 200 Absrac We show ha a planar graph wih girh a leas 20 2 has circular chromaic number a mos 2+ 1, improving earlier resuls. This follows from a general resul esablishing homomorphisms ino special arges for graphs wih given girh and given maximum average degree. Oher applicaions concern oriened chromaic number and homomorphisms ino mixed graphs wih colored edges. Keywords: graph coloring, graph homomorphism, planar graph, circular coloring, oriened coloring, discharging, -nice graphs 1 Inroducion Circular coloring, inroduced by Vince [1], is a model for coloring he verices of graphs ha provides a more refined measure of coloring difficuly han he ordinary chromaic number. A (k, d)-coloring of a graph G is a map φ: V (G) {0,..., k 1} such ha d φ(u) φ(v) k d for every edge Parially suppored by grans and of he Russian Foundaion for Fundamenal Research and he Duch-Russian Gran NWO Parially suppored by NSF gran DMS and by INTAS gran Parially suppored by NSA gran MDA The U.S. grans supporing A.V. Kosochka and D.B. Wes require he disclaimer ha any opinions, findings, and conclusions or recommendaions expressed in his publicaion are hose of he auhor(s) and do no necessarily reflec he views of he NSF or NSA. 1

2 uv E(G); a graph having such a coloring is (k, d)-colorable. Noe ha a (k, 1)-coloring is an ordinary proper k-coloring. The circular chromaic number χ c (G) of a graph G is he infimum of k over all pairs (k, d) such ha G is (k, d)-colorable. A (k, d)-coloring is d circular in he sense ha we may view he k colors as poins on a circle, and he requiremen for (k, d)-coloring is ha he colors on adjacen verices mus be a leas d posiions apar on he circle. Zhu [15] provides a horough survey of resuls on circular chromaic number. We call χ c a refined measure of coloring because χ(g) 1 < χ c (G) χ(g) for every graph G, as proved in [1] and again in [2]. A -chromaic graph is no 2-colorable, bu if is circular chromaic number is near 2, hen i is somehow jus barely no 2-colorable. For odd cycles, χ c (C 2+1 ) = If G conains C 2+1, hen χ c (G) By he heorem of Grözsch [7], every riangle-free planar graph is -colorable. In generalizing his o circular chromaic number, we ask wha hreshold on girh is needed o force he circular chromaic number o be a mos A relaxaion for planar graphs of a conjecure of Jaeger [9] on nowhere-zero flows saes he following: Conjecure 1.1 For every posiive ineger, every planar graph wih girh a leas 4 has circular chromaic number a mos When = 1, Conjecure 1.1 reduces o Grözsch s Theorem. The conjecure is sharp if rue, as shown by DeVos [5]. The example of DeVos consiss of 4 1 pahs of lengh 2 1 wih a common endpoin, plus a cycle of lengh 4 1 hrough he verices on he opposie ends of he pahs. The graph has girh 4 1, bu i has no (2 + 1, )-coloring (he color on he cenral verex canno appear on he peripheral cycle, and wih his resricion ha cycle canno be colored). Thus he circular chromaic number exceeds Nešeřil and Zhu [12] and Galuccio, Goddyn, and Hell [6] proved ha every planar graph wih girh a leas 10 4 has circular chromaic number a mos In [6] here are also analogous bounds for all surfaces. The odd girh of a graph is he minimum lengh of is odd cycles; Klosermeyer and Zhang [10] showed ha for circular chromaic number a mos i suffices o have odd girh a leas 10 4, via he Folding Lemma. Zhu [14] improved he bound from 10 4 o 8. We furher lower his hreshold o Our resul applies in a more general seing for which we inerpre (k, d)-colorings as homomorphisms. A homomorphism from G ino H is a map φ: V (G) V (H) such ha adjacen verices in G are mapped 2

3 ino adjacen verices in H. Le H(k, d) be he circulan graph wih verex se {0,..., k 1} in which i and j are adjacen if and only if d i j k d. Every (k, d)-coloring of a graph G is a homomorphism from G ino H(k, d). We acually consider a sill more general seing ha also applies o oriened homomorphisms and o homomorphisms ino mixed graphs wih colored edges and arcs. Oher applicaions include a new upper bound on he girh of planar graphs wih oriened chromaic number greaer han 5. In paricular, we show ha for every orienaion of a planar graph wih girh a leas 1, here is a homomorphism ino some ournamen wih five verices. This was known previously for girh a leas 14 via a lenghy proof [], bu he improvemen o 1 is an easy consequence of our main heorem. Our original moivaion was he resul of Zhu on Conjecure 1.1. To see how our heorem improves his hreshold from 8 o 20 2, we firs sae he relevan special case of our heorem. For graphs wih given girh and given upper bound on he average degree of all subgraphs, we prove he exisence of homomorphisms ino special arges. For his example, we are ineresed in he arge H(2 + 1, ). Theorem 1.2 If g 6 2 and d < 2 +, hen every graph wih girh g 5 2 whose subgraphs all have average degree a mos d is (2 + 1)-colorable. Corollary 1. If G is a planar graph G wih girh a leas 20 2, hen G is (2 + 1 )-colorable. Proof. Since 20 2 > 6 2, Theorem 1.2 applies if we can show ha every subgraph of G has average degree less han 2 +. I suffices o show his 5 2 bound for G iself, since every subgraph has girh a leas as large as he girh of G. We may assume ha G is 2-conneced, since we can combine (2 + 1, )- colorings of blocks. Le n, m, f be he numbers of verices, edges, and faces in some planar embedding of G. By summing face-lenghs, he bound on girh yields f 6m/(20 2). Wih d denoing he average degree, we have m = dn/2. Using Euler s Formula, we have 2 = n m+f m( 2 1+ ). d 10 1 Since he facor in parenheses mus be posiive, we obain 10 4 < 2; ha 10 1 d is, d < The argumen of Corollary 1. holds equally well for graphs embeddable on he projecive plane. For graphs embeddable on he orus or Klein bole, i also works when he girh is sricly greaer han 20 2.

4 Secion 2 inroduces definiions and noaion and saes he main resul. In Secion, we apply he resul o graphs embedded in surfaces, o circular chromaic number, o oriened chromaic number of graphs wih large girh, and o homomorphisms of planar graphs wih colored edges. We prove he main resul in Secion 4 using a discharging argumen. 2 -Expansive s-graphs Since we are ineresed in applicaions o boh graphs and digraphs, we use a model ha incorporaes boh and is sill more general. A mixed graph is a common generalizaion of muligraphs and mulidigraphs ha allows boh ordered pairs (arcs) and unordered pairs (edges) of verices in he edge se. We augmen such a srucure by allowing a fixed coloring of he edges and arcs. Definiion. Le s denoe a pair (s 1, s 2 ) of nonnegaive inegers. An s-graph is a mixed graph wihou loops in which he arcs are colored from he se { 1,..., s 1 } and he edges are colored from he se {1,..., s 2 }, in such a way ha wo edges or wo arcs do no have he same color if hey have he same wo endpoins. In he applicaions discussed in Secion, we will only consider he cases where s is (0, 1) or (1, 0) or (0, ). A (0, 1)-graph is a simple undireced graph, and a (1, 0)-graph is an orienaion of such a graph. A (1, 2)-graph has arcs of color 1 and edges of colors 1 and 2. The noion of homomorphism exends o s-graphs; a homomorphism from an s-graph G o an s-graph H is a map from V (G) ino V (H) such ha he image of an edge or forward arc wih color c in G is an edge or forward arc wih color c in H. A walk (of lengh ) in an s-graph is a lis (v 0, e 1, v 1,..., e, v ) such ha each e i is an edge or an arc, and for each i he endpoins of e i are v i 1 and v i. The edges or arcs of a walk need no be disinc, and arcs need no be followed from ail o head. A walk wih firs verex v and las verex w is a v, w-walk. The paern of a walk of lengh is he lis (C 1,..., C ), where C i is he pair (c i, σ i ) consising of he color c i of e i and a sign σ i defined by { +, if ei is undireced or v σ i = i is he head of e i,, if v i is he ail of e i. 4

5 In paricular, he elemen ( 2, ) in he paern of a walk means ha he corresponding sep is an arc wih color 2 raversed from head o ail. For an s-graph G, he paerns of lengh are he -uples (C 1,..., C ) such ha C i = (c i, σ i ) wih c i { s 1,..., 1, 1,..., s 2 } and σ i = + if c i > 0 and σ i {+, } if c i < 0. The se of paerns is deermined by s. Definiion. An s-graph G is -nice if for all v, w V (G) (no necessarily disinc) and every paern of lengh, here is a v, w-walk wih his paern. When s = (0, 1), an s-graph G is essenially jus an undireced graph, and here is only one paern of lengh. Since v, v-walks are needed, no s-graph is 1-nice. The complee graph K n is 2-nice as a (0, 1)-graph if n. No biparie graph is -nice for any, since a v, w-walk of lengh can exis only when has he same pariy as he disance from v o w. The odd cycle C 2k+1 is 2k-nice, since from a fixed verex v every verex can be reached by a pah of even lengh a mos 2k, and such a walk can be increased o lengh 2k by repeaing edges. On he oher hand, C 2k+1 is no (2k 1)-nice, because a pah of odd lengh from a verex o iself mus raverse he full cycle, and hen is lengh is a leas 2k + 1. Remark 2.1 Every -nice s-graph is also ( + 1)-nice. Proof. Le G be a -nice s-graph; noe ha 2. Given verices v and w in G and a paern (C 1,..., C +1 ), here is a walk of lengh 1 wih paern C 1 from v o some verex u, since G is -nice. Again since G is -nice, here is a u, w-walk wih paern (C 2,..., C r+1 ). Togeher, hese form a v, w-walk of lengh + 1 wih paern (C 1,..., C r+1 ). A graph is nice if i is -nice for some. Nice graphs were used implicily in [11]. They were explicily sudied and characerized in [8]. In [8] i was also shown ha minimal graphs ha are homomorphic images of all planar s-graphs wih girh a leas g are nice. For anoher example, le C 2 n denoe he direced graph whose verices are he congruence classes modulo n and whose arcs are he ordered pairs of he form (i, i + 1) and (i, i + 2). This is an s-graph for s = (1, 0). There are 2 paerns of lengh, bu o show ha C 2 n is no (n 2)-nice we only need o show failure for one paern and one pair of verices. A walk whose paern has only plus-signs follows only forward arcs. We claim ha C 2 n has no such 5

6 walk of lengh n 2 from verex 0 o verex n. The oal upward moion in n 2 seps ha each add +1 or +2 is a leas n 2 and a mos 2n 4. Since no value in his range is congruen o n modulo n, he needed walk does no exis. On he oher hand, C 2 n is (n 1)-nice. This is easies o show (see Example 2.) by using a sronger concep ha we inroduce nex. Definiion. A verex v V (H) is a (c, σ)-successor of a se W of verices in an s-graph H if for some w W here is an edge or arc wih endpoins w and v ha has color c and sign σ when viewed from w o v. An s-graph H wih n verices is -expansive if for all nonempy W V (H) and every pair (c, σ), he number of (c, σ)-successors of W is a leas min{n, W + n 1 }. A verex in W may be a (c, σ)-successor of W. In a (0, 1)-graph G, he number of successors of W is he number of verices of G having neighbors in W. No biparie graph is -expansive, because a larges parie se does no have enough successors. We formalize -expansiveness for he wo examples ha we discussed earlier and will apply laer. Example 2.2 As a (0, 1)-graph, he undireced cycle C 2k+1 is 2k-expansive. Proof. When W 2k we need only W +1 successors. The elemens of W have disinc successors in he clockwise direcion. Since he oal number of verices is odd, when W < 2k + 1 here is a verex v on he cycle such ha v / W bu he verex w ha is wo posiions laer in he clockwise direcion belongs o W. Now he common neighbor of v and w is a successor of W ha we have no already couned. Example 2. As a (1, 0)-graph, he digraph C 2 n is (n 1)-expansive. Proof. Recall ha he arcs are he pairs of he form (i, i + 1) and (i, i + 2), modulo n. Here (c, σ) can be ( 1, +) or ( 1, ), seeking ou-neighbors of W or in-neighbors of W in he digraph, respecively. By symmery, we consider only ( 1, +). As in Example 2.2, we need only W + 1 successors. For each i W, he verex i + 1 is a successor; his yields W successors. If W < n, hen here exiss i such ha i W and i + 1 / W, and now i + 2 is a successor of W ha we have no already couned. By saring wih W = {v} and using applicaions of he definiion of - expansive, i follows ha from each verex v in a -expansive s-graph, we can 6

7 reach every verex (including v) via a walk of lengh wih a specified paern. Hence every -expansive s-graph is -nice. The more deailed saemen below is wha we need from his concep; again i is immediae from he definiion of -expansive. Remark 2.4 Le v be a verex in a -expansive s-graph H wih n verices. For every paern (C 1,..., C l ) of lengh l, he number of verices w such ha H conains a v, w-walk of lengh l wih paern (C 1,..., C l ) is a leas 1 + (n 1)l/. For ordinary graphs, we say ha a graph G is H-colorable if here is a homomorphism from G ino H; his reduces o ordinary k-colorabiliy when H = K k. We use he erm H-colorable for s-graphs in he same way. When H is an s-graph, an s-graph G is criically non-h-colorable if G is no H- colorable bu every proper subgraph of G is H-colorable. The skeleon of an s-graph G is he muligraph obained from G by ignoring he orienaions of is arcs and erasing all colors. Our main resul is he following. Theorem 2.5 Le H be a -expansive s-graph. Le G be an s-graph whose skeleon has girh g and has no subgraph wih average degree more han d. If g 2 and d < 2 + 6, hen G has a homomorphism ino H. 5 4 Since every subgraph of an H-colorable s-graph is also H-colorable, an equivalen phrasing of Theorem 2.5 is ha if H is a -expansive s-graph, and G is a criically non-h-colorable s-graph whose skeleon has girh a leas 2, hen he average verex degree of he skeleon is a leas Applicaions In his secion we presen several applicaions of Theorem 2.5. We begin by considering graphs embedded on surfaces. Noe ha all graphs embeddable in he plane also embed on higher surfaces. Corollary.1 Le H be a -expansive s-graph, and le G be a criically non- H-colorable s-graph of order n. If he skeleon of G embeds on he projecive plane, hen is girh is less han If i embeds on he orus or Klein bole, hen is girh is a mos

8 Proof. Le m be he number of edges in he skeleon of G. If he claimed conclusion fails, hen he girh g of he skeleon is a leas 2, and Theorem 2.5 applies. Wih average degree a leas , we have m n Le f be he number of faces in an embedding of he skeleon on a surface of Euler characerisic N. Since fg 2m, Euler s Formula yields ( N = n m + f m ) ( 2 = m g g ). 5 1 For he surfaces menioned, N 2. Hence 2 g only when N = , wih equaliy possible We review he applicaion o circular coloring described in Corollary 1., using he more general language. Corollary.2 Le be a posiive ineger. If G is a projecive planar graph wih girh a leas 20 2, or a graph embedding on he orus or Klein bole wih girh greaer han 20, hen χ c (G) Proof. The conclusion is equivalen o he exisence of a C 2+1 -coloring of G. By Remark 2.2, C 2+1 is 2-expansive as a (0, 1)-graph. Since aking subgraphs canno reduce girh, Corollary.1 implies ha G has no subgraph ha fails o be C 2+1 -colorable. The bound of Corollary.2 generalizes easily for oher surfaces. Nex we apply Corollary.1 o oriened coloring. Definiion. The oriened chromaic number of a simple graph G is he minimum k such ha every orienaion of G admis a homomorphism ino some simple digraph wih k verices. The arge digraph can be differen for differen orienaions of G, and we may assume ha in each case he arge is an orienaion of K k. In [11,, 4], bounds on he oriened chromaic number of planar graphs wih given girh were considered. I was proved in [11] ha here are planar graphs wih arbirarily large girh having oriened chromaic number 5, and ha every planar graph wih girh a leas 16 has oriened chromaic number a mos 5. I was also menioned ha some planar graph wih girh 7 has o(7) > 5. The quesion hen is wha is he minimum hreshold g such ha all planar graphs wih girh a leas g have oriened chromaic number a 8

9 mos 5. In [], he hreshold 16 was reduced o 14 by a somewha lenghy argumen. Corollary.1 yields an immediae srenghening o 1 and applies also for hree addiional surfaces. Corollary. Every graph wih girh a leas 1 ha embeds on he orus or Klein bole has oriened chromaic number a mos 5. In fac, every orienaion of such a graph has a homomorphism ino he same 5-verex regular ournamen C5. 2 Proof. The orienaions of a simple graph G are precisely he (1, 0)-graphs wih skeleon G. If some orienaion of G is no C5-colorable, 2 hen here is a criical such digraph D, and is girh is as large as ha of G. By Remark 2., C5 2 is 4-expansive. By Corollary.1, he girh of D is a mos 40 2, which is less han 1. Le M α denoe he family of all graphs such ha he average degree of every subgraph is sricly less han α. Theorem 2 in [4] says ha every graph in M 16/7 wih girh a leas 11 admis a homomorphism ino he ocahedron K 2,2,2. Since a planar n-verex graph wih girh g has a mos (n 2) edges, i follows ha every planar graph wih girh a leas 16 has average degree less han 16/7 and hence is K 2,2,2 -colorable. The proof of Theorem 2 in [4] akes more han pages. However, a sronger version of is corollary for planar graphs follows direcly from Theorem 2.5. Corollary.4 If G is a planar graph wih girh a leas 1, hen G is K 2,2,2 - colorable. Proof. Theorem 2.5 applies, since K 2,2,2 is 5-expansive, 1 = 5 2, and = 16 when = Finally, we apply he edge-pariioning aspec of Theorem 2.5. In connecion wih quesions in group heory, Alon and Marshall [1] sudied homomorphisms of graphs wih colored edges. Given a -edge-coloring of a simple graph G, le λ(g, c) be he leas number of verices in an edge-colored graph H such ha here is a coloring-preserving homomorphism from G ino H. In our erminology, his is a homomorphism of s-graphs, where s = (0, ). Alon and Marshall discussed he maximum of λ(g, c) when G is planar and c uses a mos colors. We obain an upper bound for planar graphs wih sufficienly large girh. g g 2 9

10 Corollary.5 Le G be a planar graph wih girh a leas For every coloring of G wih a mos colors, here is a color-preserving homomorphism from G ino an edge-colored graph wih 2+1 verices. The same image always suffices: a coloring of K 2+1 in which each color class is a spanning cycle. Proof. I is well known ha he complee graph K 2+1 decomposes ino Hamilonian cycles. Le H denoe he (0, )-graph obained by coloring he ih cycle in he decomposiion of K 2+1 wih color i. We obain ha H is 2-expansive, since for each iem in a paern, we can apply Remark 2.2 o he cycle in he corresponding color. By Corollary.1, every -edge-colored planar graph wih girh a leas 20 2 is H-colorable. 4 The Main Resul We will break he proof of Theorem 2.5 ino several lemmas. Throughou his secion, fix H as a -expansive s-graph wih n verices, and le G be a criically non-h-colorable s-graph whose skeleon has girh a leas 2. We develop various properies of he skeleon o show ha is average verex degree is large. We begin by inroducing convenien noaion. Le G denoe he skeleon of G. We use δ( G) for he minimum degree of G and d(v) for he degree in G of a verex v. If d(v) = 1, hen {v} can have only one (c, σ)-successor. Since H is -expansive, we conclude ha δ( G) 2. Definiion. A hread in a graph G is a pah whose inernal verices have degree 2 in G. Two verices are weak neighbors or weakly adjacen if hey are he endpoins of a hread (his includes adjacen verices, since hreads may have no inernal verices). Our approach in proving he lemmas is as follows. If he desired conclusion fails, hen from G we delee some verices o obain a subgraph G. By he criicaliy of G, G is H-colorable. Using he -expansiveness of H, we exend he resuling homomorphism from G ino H o obain an H-coloring of G, hus producing a conradicion. Lemma 4.1 Every hread in G has lengh a mos 1. 10

11 Proof. Oherwise, le P be a u, v-hread of lengh in G, and le G be he s-graph obained from G by deleing he inernal verices of P. Since 2, G is a proper subgraph of G. Criicaliy of G yields an H-coloring φ of G. Le (C 1,..., C ) be he paern of P. By he definiion of -nice, H conains a φ(u), φ(v)-walk wih paern (C 1,..., C ). By defining φ on he inernal verices of P using he verices of his walk in succession, we exend φ o an H-coloring of G, which is impossible. Corollary 4.2 No hree verices of Y are pairwise weakly adjacen, and no wo hreads have he same se of endpoins. Proof. Oherwise, by Lemma 4.1, G has a cycle of lengh a mos. Definiion. When u and v are weakly adjacen, le l uv denoe he lengh of a shores u, v-hread. Le Y = {v V ( G): d(v) }. A weak neighbor u of v is a weak Y -neighbor if u Y ; oherwise i is a weak 2-neighbor. For v V ( G), le N Y (v) denoe he se of weak Y -neighbors of v in G. For v Y, le f(v) = + u N Y (v) ( l vu). The nex wo lemmas place lower bounds on f(v) and on u N Y (v) f(u). The moivaion for doing his is as follows. If f(v) is small, hen v has few weak Y -neighbors or long hreads o hem. Boh condiions end o reduce he average verex degree. Since we wan a lower bound on he average verex degree in G, i helps o place lower bound on values of f. Lemma 4. If v Y, hen f(v) 1. Proof. Le G be he s-graph obained from G by deleing v and all is weak 2-neighbors. By he minimaliy of G, here is an H-coloring φ of G. Suppose ha he desired inequaliy fails; we exend φ o an H-coloring of G. Consider u N Y (v). Le P u be he u, v-hread in G. Le W 0 = {u}, and for i > 0 le W i be he se of verices a which he ih verex of P could be embedded in exending φ along P u. Since H is -expansive, W i 1 + n 1 i. Leing i = l vu, we conclude ha a mos n 1 ( l vu ) verices of H are excluded from serving as he image of v in an exension of φ o P u. If some verex of H is no excluded by P u for any u N Y (v), hen φ exends o G. Hence we conclude ha n 1 ( l vu ) n > n 1, u N Y (v) 11

12 which yields u N Y (v) ( l vu) > and hence f(v) > 0. Now f(v) 1 follows from he inegraliy of f(v). Lemma 4.4 If v Y, hen u N Y (v) f(u) + 1. Proof. Say ha a verex u N Y (v) is v-free if f(u) l uv. Le G be obained from G by deleing he verex v, he v-free verices, and all heir weak 2-neighbors. By he criicaliy of G, here is an H-coloring φ of G. By Corollary 4.2, φ is defined on all of N Y (u) {v} for each u N Y (v). If u is v-free, hen as in Lemma 4. he verices of N Y (u) {v} exclude a mos n 1 y N Y (u) {v} ( l yu) verices of H from serving as he image of u in an exension of φ along he hreads from N Y (u) {v} o u. By he definiion of f, his quaniy equals n 1 (f(u)+l vu ). Since u is v-free, we have f(u)+l vu, and he number of verices excluded from serving as he image of u is a mos n 1. In paricular, a leas one is available. Accouning for he possibiliy ha f(u) + l vu is sricly less han, we compue ha a leas 1 + n 1 ( l vu f(u)) verices of H are available for he image of u. In exending he homomorphism along he hread from u o v, he number of possible images for he curren verex increases by a leas n 1 wih each of he l vu seps, because H is -expansive. By he ime we reach v, here are hus a leas 1 + n 1 ( f(u)) verices available for he image of v in an exension of φ o all he weak neighbors of u. The number of verices forbidden from being he image is a mos n 1 n 1 ( f(u)), which equals n 1 f(u). When u N Y (v) is no v-free, φ(u) is fixed. In his case, as in Lemma 4., a leas 1 + n 1 l vu verices of H can serve as he image of v in an exension of φ o he u, v-hread, and a mos n 1 ( l vu ) verices are excluded. Since l vu < f(u), fewer han n 1 f(u) verices are excluded. Since G has no H-coloring, every verex of H mus be excluded from serving as he image of v in a leas one of hese exensions. This requires n 1 u N Y (v) f(u) n. Since every f(u) is an ineger, we have f(u) + 1. u N Y (v) We complee he proof using a discharging argumen. We rea d(v) as an iniial charge on he verex v V ( G). We will move charge from verex o verex, wihou changing he oal, o obain a new charge d (v) such ha d (v) 2 + 4d(v) for all v V ( G). (1)

13 Le p = V ( G) and m = E( G). If (1) holds, hen 2m = d (v) ( 2 + 4d(v) 2 ) = 5 v V ( G) v V ( G) ( ) 2p m, and hence 5 1 p 5 4 m. This makes he average degree of G a leas 2(5 1) 5 5 which equals Hence i suffices o obain 5 4 d so ha (1) holds. 5 4, Discharging Rules Given a muligraph G wih d(u) denoing he degree of verex u as an iniial charge, define an adjused charge d (u) for each u V ( G) by he following operaions: R1. Every v Y gives each weak 2-neighbor he amoun 5. R2. Every v Y gives each weak Y -neighbor he amoun f(v)+(+1)(d(v) ) 5d(v). Lemma 4.5 Every v Y receives from is weak Y -neighbors a leas Proof. If every u N Y (v) sends v a leas f(u), hen v receives from N 5 Y (v) a leas 1 5 u N Y (v) f(u). By Lemma 4.4, u N Y (v) f(u) + 1. Hence we may assume ha f(u)+(+1)(d(u) ) < f(u) for some u N 5d(u) 5 Y (v). This requires d(u) 4. Hence we can cancel d(u) afer clearing fracions o obain f(u) > + 1. When f(u) + 1, he formula in R2 yields ha u by iself gives v a leas +1. For y N 5 Y (v), we have d(y) and f(y) 1 (by Lemma 4.), so all oher amouns sen o v are nonnegaive. Lemma 4.6 The inequaliy (1) holds for he charge d obained for G via he Discharging Rules. Proof. If d(v) = 2, hen v sends ou nohing and receives from each of 5 is wo weak Y -neighbors. Thus d (v) = = 2 + 4d(v) Now consider v Y. Verex v sends ou 5 w N Y (v) (l vw 1) o is weak 2-neighbors and f(v)+(+1)(d(v) ) o is weak Y -neighbors. By Lemma 4.5, v 5 also receives a leas +1 from is weak Y -neighbors. Since also f(v) = + 5 ( lvw ) by definiion, we obain (wih each sum running over w N Y (v)) d (v) d(v) f(v) + ( + 1)(d(v) 4) (lvw 1) 5 5 = d(v) [ + ] ( l wv + l vw 1) 5 = d(v) 5 [5 ( 1) ( + 1)] ( + 1)(d(v) 4)) 5 ( + 2)d(v) [ + 4( + 1)] =. 5

14 Since d(v) and 2, we have ( + 2)d(v) = ( + 2)d(v) ( 2) + (10 2) = (d(v) ) + 2d(v) (10 2) 4d(v) Therefore, (+2)d(v) and also he heorem. 2+ 4d(v) 2, which complees he proof of he lemma 5 References [1] N. Alon and T.H. Marshall, Homomorphisms of edge-coloured graphs and Coxeer groups. J. Algebraic Combinaorics 8 (1998), 5 1. [2] J. A. Bondy and P. Hell, A noe on he sar chromaic number. J. Graph Theory 14 (1990), [] O. V. Borodin, A. V. Kosochka, J. Nešeřil, A. Raspaud, and E. Sopena, On he maximum average degree and he oriened chromaic number of a graph, Discree Mah. 206 (1999), [4] O. V. Borodin, A. V. Kosochka, J. Nešeřil, A. Raspaud, and E. Sopena, On universal graphs for planar oriened graphs of a given girh, Discree Mah. 188 (1998), [5] M. DeVos, communicaion a Workshop on Flows and Cycles, Simon Fraser Univ., June [6] A. Galuccio, L. Goddyn, and P. Hell, High girh graphs avoiding a minor are nearly biparie, J. Comb. Theory (B) 8 (2001), [7] H. Grözsch, Ein Dreifarbensaz für dreikreisfreie Neze auf der Kugel, Wiss. Z. Marin-Luher-U., Halle-Wienberg, Mah.-Na. Reihe 8 (1959), [8] P. Hell, A. V. Kosochka, A. Raspaud, and E. Sopena, On nice graphs, Discree Mah. 24 (2001), [9] F. Jaeger, On circular flows in graphs, in: Finie and Infinie Ses (Eger, 1981), Colloq. Mah. Soc. J. Bolyai 7 (1984), Norh Holland,

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