A note on degenerate and pectrally degenerate graph Noga Alon Abtract A graph G i called pectrally d-degenerate if the larget eigenvalue of each ubgraph of it with maximum degree D i at mot dd. We prove that for every contant M there i a graph with minimum degree M which i pectrally 50-degenerate. Thi ettle a problem of Dvořák and Mohar. Introduction The pectral radiu ρg) of a finite, imple) graph G i the larget eigenvalue of it adjacency matrix. A graph i d-degenerate if any ubgraph of it contain a vertex of degree at mot d. A reult of Haye [5] aert that any d-degenerate graph with maximum degree at mot D ha pectral radiu at mot dd. In fact, the reult i a bit tronger, a follow. Propoition. [5]) Let G be a graph having an orientation in which every outdegree i at mot d and every indegree i at mot D. Then ρg) dd. For completene we include a imple proof which i omewhat horter than the one given in [5]). Proof: Denote the vertice of G by {,,..., n}, and let d + i and d i be the indegree and outdegree of vertex i in an orientation of G in which every outdegree i at mot d and every indegree i at mot D. The pectral radiu of G i the maximum poible value of the um ij E x ix j where E denote the et of oriented edge of G and the maximum i taken over all vector x, x,..., x n ) atifying i x i =. For each oriented edge ij E, x i x j dd x ix j d + i d j x i d + i + x j d j. The deired reult now follow by umming over all oriented edge. Following Dvořák and Mohar [3], call a graph G pectrally d-degenerate, if for every ubgraph H of G, ρh) ddh), where DH) i the maximum degree of H. Thu, by Propoition., Sackler School of Mathematic and Blavatnik School of Computer Science, Tel Aviv Univerity, Tel Aviv 69978, Irael and Intitute for Advanced Study, Princeton, New Jerey, 08540. Email: nogaa@tau.ac.il. Reearch upported in part by an ERC Advanced grant, by a USA-Iraeli BSF grant, by the Owald Veblen Fund and by the Bell Companie Fellowhip.
every d-degenerate graph i pectrally 4d-degenerate. The author of [3] proved the following rough convere: *) Any pectrally d-degenerate graph with maximum degree at mot D d contain a vertex of degree at mot 4d log D/d). They further howed that the dependence on D cannot be eliminated if the dependence on d i ubexponential and aked whether there i a function f mapping poitive integer to poitive integer uch that for every d, any pectrally d-degenerate graph contain a vertex of degree at mot fd). In thi note we ettle thi problem and how that there i no uch function by proving the following. Theorem. For every poitive integer M there i a pectrally 50-degenerate graph G in which every degree i at leat M. Our proof combine the approach of [3], which i baed on a contruction of [6], with ome additional probabilitic argument. The contant 50 can be reduced, and we make no attempt to optimize it, or the other abolute contant that appear in the proof. To implify the preentation, we omit all floor and ceiling ign whenever thee are not crucial. Spectrally degenerate graph of high degree. A probabilitic contruction In thi ubection we decribe a probabilitic contruction which i imilar to the one given in [6]. Theorem. For every poitive integer M and all ufficiently large n > n 0 M) there exit a bipartite graph G with vertex clae A and B, atifying the following propertie. i) B A = n. ii) Every vertex of A ha degree M and every vertex of B ha degree larger than 000M. iii) Every ubgraph of G with average degree at leat 0 contain a vertex of degree at leat 000M. Proof: Fix ɛ > 0, ɛ < 4 M. Let B be a dijoint union of M et B,..., B M, where B i = n 4iɛ. The graph G i a random graph contructed a follow. For each i, i M, let G i be a bipartite graph on the vertex clae A and B i coniting of a random et of A edge obtained by picking, for each a A, a uniform random b B i, taking ab to be an edge of G i. The graph G conit of all edge of all graph G i. Note that the degree of every vertex of G that lie in A i exactly M, and that if n i ufficiently large then the degree of every vertex of G that lie in B i, aymptotically almot urely a.a.., for hort), at leat n 4ɛ / > 000M, provided n i ufficiently large. Here we ay that a property hold a.a.. if the probability it hold tend to a n tend to infinity, and the claim about the degree follow eaily from the known etimate for binomial ditribution, a the degree of each vertex in B i i a binomial random variable with parameter n and / B i whoe expectation i n 4iɛ.
Therefore the graph G atifie the propertie i) and ii) a.a.. It remain to how that it atifie property iii) a well a.a.. Let H be a ubgraph of G with average degree at leat 0, and let 0) denote the number of it vertice. We conider three poible cae. Cae : B M = n 4M ɛ. We how that in thi cae, a.a.., the graph G contain no ubgraph on vertice with average degree at leat 0. Indeed, uch a ubgraph can contain at mot edge incident with vertice of B M a each A-vertex ha only one neighbor in B M ) and hence there are at leat 4 edge of H incident with vertice in i<m B i. But the probability that there i uch a collection of 4 edge i at mot ) n ) ) 4 B M )4 en ) e 8 )4 B M )4 c n 3 B M )4, where c = e)e/8) 4 i an abolute contant. The lat quantity i at mot Therefore, a.a.., there i no uch H. Cae : B = n 4ɛ. c n n 4M ɛ ) 3 n 4M ɛ ) 4 = c n 3 4M +4 M )ɛ < n ɛ. In thi cae the graph ha at leat 4 edge incident with vertice in i B i. A the total ize of thee et i maller than n ɛ Cae 3: There i an i, i < M, o that the graph mut have degree at leat n ɛ > 000M, a needed. n 4i+ɛ = B i+ < B i = n 4iɛ. In thi cae the graph H ha at mot edge incident with vertice in B i B i+. If it maximum degree doe not exceed 000M, it ha at mot 000M j i+ B j = o) edge incident with vertice in j i+ B j. It thu ha at leat 3 o)) edge incident with vertice in j<i B j. The probability P that there i uch a collection of edge can be bounded a in the firt cae. Indeed, P i at mot the following, where in all um over, the parameter range over all value between B i+ and B i. P ) n ) ) 3 o)) B i )3 o) 3
c n o)) n 4i ɛ ) 3 o)), for an appropriate abolute contant c. A B i in the above range, the lat quantity i at mot c n + 4i ɛ) o)) 4 i ɛ)3 o)) ) = c n o) 4i ɛ+3 4 i ɛ) < n ɛ. Thi how that the event in Cae 3 alo doe not occur a.a.., and complete the proof of the theorem.. Bounding the pectral degeneracy In order to prove Theorem. we need the following lemma, which i a bit tronger than a imilar lemma proved in [3]. Lemma. Suppoe M M 0, where M 0 i a large integer. Let H be a bipartite graph with vertex clae A and B in which every vertex of A ha degree at leat 0 and at mot M, and every vertex of B ha degree at leat D/00 and at mot D, where D 000M. Then there i a ubet A A o that the induced ubgraph of H on A B ha minimum degree at leat 0 and maximum degree maller than 000M. Proof: It uffice to how that H contain an induced ubgraph on A B, where A A, in which the degree of every vertex of A i at leat 0, and the degree of every vertex of B i at leat M > 0) and at mot 800M. We proceed to prove thi tatement uing the Lová Local Lemma proved in [4] c.f. alo, e.g., []). Our approach follow the one in []. Starting with H 0 = H, contruct a equence H i = V i, E i ) of induced ubgraph of H, where for each i, H i i a random induced ubgraph of H i with vertex clae A i and B, where A i i obtained by picking each vertex in A i, randomly and independently, to lie in A i, with probability /. Note that by contruction, the degree of each vertex of A i in H i i exactly it degree in H, which, by aumption, i at leat 0. For each vertex v B, let d i v) denote the degree of v in H i. We claim that with poitive probability, a long a D/ i 300M, then d i v) i cloe to d 0 v)/ i. More preciely, let j be the maximum value of i o that D/ i 300M. Note that D/ j < 600M. Define a equence a 0 v) = d 0 v) and a i v) = a i v) + a i v) /3 for all i, 0 < i j, and imilarly a equence b 0 v) = d 0 v) and b i v) = b i v) b i v) /3. Then, with poitive probability D 00 i 0.9 b 0v) i + r i for all i j and all v B. b r v) /3 ) = b iv) d i v) a i v) = a 0v) i r i a r v) /3 ) D i. Indeed, the above tatement i proved by induction on i. For i = 0 there i nothing to prove. Auming the above hold for i, we etablih the aertion for i uing the Local Lemma. To do o conider, for each vertex v B, the event F v that d i v) fail to atify d i v) d i v) /3 d i v) d i v) 4 + d i v) /3.
Each event F v i mutually independent of all other event F u beide thoe for which u and v have a common neighbor in H i. There are clearly at mot d i v)m < d i v) uch vertice u, and hence the fact that d i v) > M, the known tandard etimate of Binomial ditribution c.f., e.g., []) and the Local Lemma imply that with poitive probability none of the event F v hold. The graph H j atifie the concluion of the lemma, completing it proof. We are now ready to prove the following reult, which clearly implie Theorem.. Theorem.3 Let G be a graph atifying the aertion of Theorem., where M > M 0 and M 0 i a in Lemma.. Then G i pectrally 50-degenerate and ha minimum degree M. Proof: Aume thi i fale and let H be a ubgraph of G with maximum degree D and pectral radiu ρh) > 50D, where D i a mall a poible. Clearly H cannot be 0-degenerate, ince otherwie ρh) < 0D < 50D and thu it contain a ubgraph with minimum degree exceeding 0, implying, by Theorem., part iii), that it maximum degree, and hence alo D, are at leat 000M. hold: We claim that there i a coloring of the edge of H by color, red and green, o that the following a) In the graph H r coniting of all red edge, the degree of every vertex of A i at mot 0. b) In the graph H g coniting of all green edge, the degree of every vertex of B i at mot D/00. To prove the claim, conider the following proce of coloring vertice and edge of H by red and green. Starting with no colored vertex or edge, repeat the following two tep a long a it i poible to apply any of them: ) If there i a yet uncolored vertex v of B incident with at mot D/00 uncolored edge, color it green, and color all uncolored edge incident with it green. ) If there i a yet uncolored vertex u of A incident with at mot 0 uncolored edge, color it red, and color all uncolored edge incident with it red. Note that only vertice of B can be green, and only vertice of A can be red. Moreover, once a vertex i colored, all edge incident with it get color a well. In addition, a vertex of B ha no green edge incident with it before it i colored, and a vertex of A ha no red edge incident with it before it i colored. The contruction thu implie that the degree of every vertex of B in the green graph i at mot D/00, wherea the degree of every vertex of A in the red graph i at mot 0. The proce terminate when either all edge are colored, or there are yet uncolored edge and hence alo vertice) and we cannot apply the rule ) and ) anymore. But if thi i the cae then in the induced ubgraph on the yet uncolored vertice every vertex of B ha degree exceeding D/00 and at mot D) and every vertex of A ha degree exceeding 0 and at mot M). By Lemma. thi contradict the fact that G atifie Theorem.. Thu we have colored all edge, a claimed. By the minimality of D, the graph H g whoe maximum degree i at mot D/00 i pectrally 50- degenerate, and by Propoition. the graph H R, which i 0-degenerate, ha maximum eigenvalue 5
at mot 0D. It follow that the pectral radiu of H i at mot 50D/00 + 0D < 50D, contradicting the choice of H. Thi complete the proof. Reference [] N. Alon, The trong chromatic number of a graph, Random Structure and Algorithm 3 99), 7. [] N. Alon and J. H. Spencer, The Probabilitic Method, Third Edition, Wiley, 008, xv+35 pp. [3] Z. Dvořák and B. Mohar, Spectrally degenerate graph: Hereditary cae, arxiv: 00.3367 [4] P. Erdő and L. Lováz, Problem and reult on 3-chromatic hypergraph and ome related quetion, in: Infinite and Finite Set A. Hajnal et. al. ed), Colloq. Math. Soc. J. Bolyai, North Holland, Amterdam, 975, pp. 609 67. [5] T. P. Haye, A imple condition implying rapid mixing of ingle-ite dynamic on pin ytem, in Proceeding of the 47th Annual IEEE Sympoium on Foundation of Computer Science FOCS 006), IEEE, 006, pp. 39 46. [6] L. Pyber, V. Rödl and E. Szemerédi, Dene graph without 3-regular ubgraph, J. Combin. Theory Ser. B 63 995), 4 54. 6