279 research outputs found

    On the Minimum Order of Extremal Graphs to have a Prescribed Girth

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    We show that any n‐vertex extremal graph G without cycles of length at most k has girth exactly k+1k+1 if k≄6k\ge 6 and n>(2(k−2)k−2+k−5)/(k−3)n>(2(k-2)^{k-2}+k-5)/(k-3). This result provides an improvement of the asymptotical known result by Lazebnik and Wang [J. Graph Theory, 26 (1997), pp. 147–153] who proved that the girth is exactly k+1k+1 if k≄12k\ge 12 and n≄2a2+a+1kan\ge 2^{a^2+a+1}k^a, where a=k−3−⌊(k−2)/4⌋a=k-3-\lfloor(k-2)/4\rfloor. Moreover, we prove that the girth of G is at most k+2k+2 if n>(2(t−2)k−2+t−5)/(t−3)n>(2(t-2)^{k-2}+t-5)/(t-3), where t=⌈(k+1)/2⌉≄4t=\lceil (k+1)/2\rceil\ge 4. In general, for k≄5k\ge 5 we show that the girth of G is at most 2k−42k-4 if n≄2k−2n\ge 2k-2

    Distance colouring without one cycle length

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    We consider distance colourings in graphs of maximum degree at most dd and how excluding one fixed cycle length ℓ\ell affects the number of colours required as d→∞d\to\infty. For vertex-colouring and t≄1t\ge 1, if any two distinct vertices connected by a path of at most tt edges are required to be coloured differently, then a reduction by a logarithmic (in dd) factor against the trivial bound O(dt)O(d^t) can be obtained by excluding an odd cycle length ℓ≄3t\ell \ge 3t if tt is odd or by excluding an even cycle length ℓ≄2t+2\ell \ge 2t+2. For edge-colouring and t≄2t\ge 2, if any two distinct edges connected by a path of fewer than tt edges are required to be coloured differently, then excluding an even cycle length ℓ≄2t\ell \ge 2t is sufficient for a logarithmic factor reduction. For t≄2t\ge 2, neither of the above statements are possible for other parity combinations of ℓ\ell and tt. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur

    The history of degenerate (bipartite) extremal graph problems

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    This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

    Many TT copies in HH-free graphs

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    For two graphs TT and HH with no isolated vertices and for an integer nn, let ex(n,T,H)ex(n,T,H) denote the maximum possible number of copies of TT in an HH-free graph on nn vertices. The study of this function when T=K2T=K_2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) ex(n,K3,C5)≀(1+o(1))32n3/2,ex(n,K_3,C_5) \leq (1+o(1)) \frac{\sqrt 3}{2} n^{3/2}, (ii) For any fixed mm, s≄2m−2s \geq 2m-2 and t≄(s−1)!+1t \geq (s-1)!+1 , ex(n,Km,Ks,t)=Θ(nm−(m2)/s)ex(n,K_m,K_{s,t})=\Theta(n^{m-\binom{m}{2}/s}) and (iii) For any two trees HH and TT, ex(n,T,H)=Θ(nm)ex(n,T,H) =\Theta (n^m) where m=m(T,H)m=m(T,H) is an integer depending on HH and TT (its precise definition is given in Section 1). The first result improves (slightly) an estimate of Bollob\'as and Gy\H{o}ri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques

    MAXIMISING THE NUMBER OF CONNECTED INDUCED SUBGRAPHS OF UNICYCLIC GRAPHS

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    Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g and k pendent vertices. In this paper, we give a partial characterisation of the structure of all maximal graphs in G(n, d, g, k) for the number of connected induced subgraphs. For the special case d = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the maximum number of connected induced subgraphs given: (1) order, girth, and number of pendent vertices; (2) order and girth; (3) order
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