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Register a new userReconstructing the degree sequence of a sparse graph from a partial deck
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The deck of a graph $G$ is the multiset of cards ${G-v:vin V(G)}$. Myrvold (1992) showed that the degree sequence of a graph on $ngeq7$ vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree $d$ can reconstructed from any deck missing $O(n/d^3)$ cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.
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Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give a sharp upper bound for rho(G) by a function of d_1, d_2, ..., d_L, which generalizes a series of previous results.
We prove an asymptotic formula for the number of orientations with given out-degree (score) sequence for a graph $G$. The graph $G$ is assumed to have average degrees at least $n^{1/3 + varepsilon}$ for some $varepsilon > 0$, and to have strong mixing properties, while the maximum imbalance (out-degree minus in-degree) of the orientation should be not too large. Our enumeration results have applications to the study of subdigraph occurrences in random orientations with given imbalance sequence. As one step of our calculation, we obtain new bounds for the maximum likelihood estimators for the Bradley-Terry model of paired comparisons.
A fundamental result of Kuhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.
The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of H`{a}n, Person and Schacht who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an $n$-vertex $3$-graph is $left(frac{5}{9}+o(1)right)binom{n}{2}$. In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of H`{a}n, Person and Schacht, and additionally allow one third of the vertices to have degree $frac{1}{9}binom{n}{2}$ below this threshold. Furthermore, we show that this result is, in some sense, tight.
The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $mathscr D$ is $1+max mathscr D$. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set $mathscr D$. We expand on their results, and determine the least size of graphs with degree set $mathscr D$ when (i) $min mathscr D mid d$ for each $d in mathscr D$; (ii) $min mathscr D=2$; (iii) $mathscr D={m,m+1,ldots,n}$. In addition, given any $mathscr D$, we produce a graph $G$ whose size is within $min mathscr D$ of the optimal size, giving a $big(1+frac{2}{d_1+1})$-approximation, where $d_1=max mathscr D$.
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