4,915 research outputs found

    An extremal problem on potentially Kmβˆ’PkK_{m}-P_{k}-graphic sequences

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    A sequence SS is potentially Kmβˆ’PkK_{m}-P_{k} graphical if it has a realization containing a Kmβˆ’PkK_{m}-P_{k} as a subgraph. Let Οƒ(Kmβˆ’Pk,n)\sigma(K_{m}-P_{k}, n) denote the smallest degree sum such that every nn-term graphical sequence SS with Οƒ(S)β‰₯Οƒ(Kmβˆ’Pk,n)\sigma(S)\geq \sigma(K_{m}-P_{k}, n) is potentially Kmβˆ’PkK_{m}-P_{k} graphical. In this paper, we prove that Οƒ(Kmβˆ’Pk,n)β‰₯(2mβˆ’6)nβˆ’(mβˆ’3)(mβˆ’2)+2,\sigma (K_{m}-P_{k}, n)\geq (2m-6)n-(m-3)(m-2)+2, for nβ‰₯mβ‰₯k+1β‰₯4.n \geq m \geq k+1\geq 4. We conjecture that equality holds for nβ‰₯mβ‰₯k+1β‰₯4.n \geq m \geq k+1\geq 4. We prove that this conjecture is true for m=k+1=5m=k+1=5 and m=k+2=5m=k+2=5.Comment: 5 page

    An extremal problem on potentially Kp,1,1K_{p,1,1}-graphic sequences

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    A sequence SS is potentially Kp,1,1K_{p,1,1} graphical if it has a realization containing a Kp,1,1K_{p,1,1} as a subgraph, where Kp,1,1K_{p,1,1} is a complete 3-partite graph with partition sizes p,1,1p,1,1. Let Οƒ(Kp,1,1,n)\sigma(K_{p,1,1}, n) denote the smallest degree sum such that every nn-term graphical sequence SS with Οƒ(S)β‰₯Οƒ(Kp,1,1,n)\sigma(S)\geq \sigma(K_{p,1,1}, n) is potentially Kp,1,1K_{p,1,1} graphical. In this paper, we prove that Οƒ(Kp,1,1,n)β‰₯2[((p+1)(nβˆ’1)+2)/2]\sigma (K_{p,1,1}, n)\geq 2[((p+1)(n-1)+2)/2] for nβ‰₯p+2.n \geq p+2. We conjecture that equality holds for nβ‰₯2p+4.n \geq 2p+4. We prove that this conjecture is true for p=3p=3.Comment: 5 page
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