# $$l_p$$ -Optimal Rankings and Max-Optimal Rankings are Different

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The colouring number col(G) of a graph G is the smallest integer Îº for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than Îº -- 1 in the remaining graph is removed at any step. An edge e of a graph G...

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Let G be a finite graph with p vertices and q edges. A vertex magic total labeling is a bijection f from V(G)âˆªE(G) to the consecutive integers 1, 2, ..., p+q with the property that for every uâˆŠV(G), f(u)+ Î£vâˆŠN(u)f(uv) =k for some constant k. Such a labeling is E-super if f...

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A graph G is called a fractional (k; n ' ;m)-critical deleted graph if any n ' vertices are removed from G the resulting graph is a fractional (k;m)-deleted graph. In this paper, we prove that for integers for each pair of non-adjacent vertices x, y of G, then G is a fractional (k; n '...

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The set of all non-increasing non-negative integer sequences Ï€ = (d1, d2,...,dn) is denoted by NSn. A sequence Ï€ Îµ NSn is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of Ï€. The set of all graphic...

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Let $$k\ge 1$$ be an integer, and let $$G$$ be a finite and simple graph with vertex set $$V(G)$$ . A signed Roman $$k$$ -dominating function (SRkDF) on a graph $$G$$ is a function $$f:V(G)\rightarrow \{-1,1,2\}$$ satisfying the conditions that (i) $$\sum _{x\in N[v]}f(x)\ge k$$ for each vertex...

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A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set {0,1, 2,..., q} such that, when each edge xy is assigned the label |f(x) -- f(y)|, the resulting edge labels are distinct. A graph G is said to be one modulo N graceful...

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For an integer $$\ell $$ at least three, we prove that every Hamiltonian $$P_\ell $$ -free graph $$G$$ on $$n>\ell $$ vertices has cycles of at least $$\frac{2}{\ell }n-1$$ different lengths. For small values of $$\ell $$ , we can improve the bound as follows. If $$4\le \ell \le 7$$ , then $$G$$...