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3 2. 3. Study on Information Processing Letters 92 5 (2004) An extension of TYT algorithm for Formula Not Shown using precomputation Formula Not Shown GF ((2 n ) m ) Formula Not Shown [9] On Edge-Forwarding Index of Graphs Edge-Forwarding Index of Graphs Bouabdallah Sotteau The Proof of a Conjecture of Bouabdallah and Sotteau Networks, 45(2) (2005) Graphs, branchwidth, and tangles! oh my! Running Headline 2-restricted edge-connectivity of vertex-transitive graphs Running Headline Connectivity of Vertex-Transitive Graphs \thanks {The work was supported by NNSF of China (No ).} NNSF Nation Natural Science Fundation 4 Jun-Ming Xu J.-M. Xu Junming Xu J. Xu J. Xu J. Xu Journal of Combinatorial Theory 3 Jun-Ming Xu J.-M. Xu Xu Junming Junming Reinhard 3

4 Diestel: Graph Theory /O Diestel Reinhard, D.Reinhard XU Junming Department of Mathematics University of Science and Technology of China Hefei, Anhui, , China Fax Fax \footnote{corresponding author: IEEE words 150 Kautz UK(d, n) λ (UK(d, n)) 4

5 Theorem For the Kautz undirected graph UK(d, n) with d 2 and n 1,, for n = 1, d = 2; 3, for n = d = 2; λ (UK(d, n)) = 2d 2, for n = 1, d 3; 4d 4, for n 2, d 3 or n 3, d 2. A graph G is λ -optimal if its restricted edge-connectivity is equal to its minimum edgedegree. This paper proves that for any d and n with d 2 and n 1 the Kautz undirected graph UK(d, n) is λ -optimal except UK(2, 1) and UK(2, 2). so-and-so conjecture so-and-so result [9] Bouabdallah and Sotteau [2] proposed the following conjecture A. Bouabdallah and D. Sotteau proposed the following conjecture in [On the edge forwarding index problem for small graphs, Networks 23 (1993), ] [10] As a consequence, we obtain the result of Hellwig, Rautenbach and Volkmann [4] that κ(l(g)) = λ (G) As a consequence, we obtain the result of Hellwig, Rautenbach and Volkmann [Information Processing Letters, 91(2004), 7-10] that κ(l(g)) = λ (G) Networks Information Processing Letters Mathematical Review Zentralblatt Math Related links Related links to the major AMS 2000 Mathematics Subject Classification 05C O157.5 O

6 6 We follow [ ] for graph-theoretical terminology and notation not defined here A short proof of so-and-so theorem, Proof of so-and-so conjecture J. Graph Theory [1, 8] 1 2 [3] [1] Archdeacon et al [1] error not true It would seem to have a flaw in their proof the result seems not true The rest of the 6

7 paper is organized as follows. In Section 2 we present some definitions and Lemmas. Our main results are in Section 3 and Section 4, respectively Claim ( ) 4. The proof proceeds by induction on n 3 We proceed by contradiction Suppose to the contrary that G k We prove the theorem by constructing k pairwise internally disjoint (x, y)-paths between any two vertices in G In order to prove the theorem, we only need to prove that To prove the theorem, it is sufficient to prove that To the end, let First, Secondly, Thirdly,, Lastly It remains for us to show that 7

8 1. Case 1, Case 2, Lemma Claim 3. the proof is here omitted for details [7] Latex [5] 5n 1 (G) + 4n 2 (G) + + n 5 (G) n 7 (G) + 2n 8 (G) + + ( 6)n (G) cr(G) (7.1) n 7 (G) + 2n 8 (G) + + ( 6)n (G) + 2 > 0. 5n 1 (G) + 4n 2 (G) + + n 5 (G) > 0 by (7.1) (7.1) 5n 1 (G) + 4n 2 (G) + + n 5 (G) n 7 (G) + 2n 8 (G) + + ( 6)n (G) cr(G) n 7 (G) + 2n 8 (G) + + ( 6)n (G) + 2 > 0, that is, 5n 1 (G) + 4n 2 (G) + + n 5 (G) > 0. (7.2) by (7.2) 8

9 8 9 The authors would like to thank the anonymous referees for their helpful comments and suggestions which considerably improved the presentation of the paper. 10 to appear in 9

10 Mathematical Review J. A. Bondy; J.-M. Xu. books [1], articles in journals [2], papers in a contributed volume [3,4], unpublished papers [5]. 1. M. Xu, X.-M Hou and J.-M. Xu, The proof of a conjecture of Bouabdallah and Sotteau. Networks, 44 (4) (2004), J.-M. Xu, Theory and Application of Graphs. Kluwer Academic Publishers, Dordrecht/ Boston/ London, C.C. Lindner and C.A. Rodger, Decomposition into cycles II, Cycle Systems in Contemporary Design Theory: A Collection of Surveys, J.H. Dinitz and D.R. Stinson (Editors), Wiley, New York, 1992, pp M. Li, Lower bounds by Kolmogorov complexity, in: Proc. ICALP 85, Lecture Notes in Computer Science, Vol. 194, Springer, Berlin, 1985, pp M. Xu, The forwarding index of networks, Ph.D. Thesis, Department of Mathematica, University of Science and Technology of China, Latex 10 Journal of Interconnection Networks Guidelines for Contributors References are to be listed in the order cited in the text [J] [M] [C] Latex Y.-C. Chen, J. J. M. Tan, L.-H. Hsu and S.-S. Kao 2003 \bibitem{cthk03}

11 Instructions to Authors Papers submitted should be written in an elegant and clear style. Prospective authors not expert in the language used in their papers are urged to seek proper assistance in this matter before submitting papers. If the language used in a submitted manuscript is poor, the manuscript will be immediately sent back to the author. Mathematical Review (2003j: 05071) [6] Finally, I must remark on the clarity of the paper s exposition and the strong concatenation of the notions in the proofs, which are very difficult in their technicality. I would say that this paper is the most beautiful that has appeared in the last 3 years in the Journal of Combinatorial Theory, Series B. It could in fact be used as a writing model for other mathematicians. 12 x01, x02, x03, tex x01-00, x01-01, x01-02, 11

12 Suppose Assume that It s doesn t The theorem is Ok The theorem follows The proof of the theorem is complete is completed A is a set of arcs G is a graph Let A be a set of arcs Let G be a graph The symbol A is a set of arcs The symbol G denotes a graph in the upper left-hand corner the second-last vertex every second edge k pairwise internally disjoint paths k 15 cm 23 cm 11 pt hard copy double spaced \maketitle \setlength{\baselineskip}{24pt} e E(H) 1 ( re(h) dn p re(h) 2 ) = (d + 1) p d p + 1 e E(H) ( 1 re(h) ) re(h) 2 = d n p (d + 1) p d p + 1 Latex.ctx \documentclass[11pt]{cctart}.ctx.tex \documentclass[11pt,cjk]{cctart}.tex CJK \renewcommand{\refname}{references} 12

13 References $ $ \, PDF camera-ready [1] D. Archdeacon, J. Ellis-Monaghan, D. Fisher, D. Froncek, P. C. B. Lam, S. Seager, B. Wei, R. Yuster, Some remarks on domination. J. Graph Theory, 46(3) (2004), [2] A. Bouabdallah and D. Sotteau, On the edge forwarding index problem for small graphs, Networks, 23 (1993), [3] Y.-M. Fan, J.-M. Xu and M. Lü, The restricted edge-connectivity of Kautz undirected graphs. Ars Combinatoria, to appear. [4] A. Hellwig, D. Rautenbach and L. Volkmann, Note on the connectivity of line graphs. Information Processing Letters, 91(1) (2004), [5] X.-M Hou, M. Xu and J.-M. Xu, Forwarding indices of folded n-cubes. Discrete Applied Math. 145 (3) (2005), [6] H. Li, On a conjecture of Woodall. J. Combin. Theory, Ser. B 86 (1) (2002), [7] Q. Li, J.-M. Xu, and Z.-L. Zhang, The infinite families of optimal double loop networks (in Chinese). Science in China, 23 A(9) (1993), [8] R. Rizzi, A short proof of Köning s matching theory. J. Graph Theory, 33 (2000), [9] M. Xu, X.-X. Hou and J.-M. Xu, The Proof of a conjecture of Bouabdallah and Sotteau. Networks, 44 (4) (2004), [10] J.-M. Xu, M. Lü, M.-J. Ma and A. Hellwig, Super connectivity of line graphs. Information Processing Letters, 94 (4) (2005),

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Erdös ? 1 2 1 1 Euler Menger Kuratowski Ramsey Whitney Dirac König Tutte Erdös Menger Kuratowski Ramsey 1 Erdös 2 1 2 1? 1 2 2 3 5 5 4 2 1 3 Haray [7] Haray 2 3 2 4 1. Mathematical Review Zentralblatt MATH http://www.ams.org/mathscinet