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Register a new userA simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs
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One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.
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Posas theorem states that any graph $G$ whose degree sequence $d_1 le ldots le d_n$ satisfies $d_i ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e. we prove a `resilience version of Posas theorem: if $pn ge C log n$ and the $i$-th vertex degree (ordered increasingly) of $G subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Diracs theorem obtained by Lee and Sudakov. Chvatals theorem generalises Posas theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvatals theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.
Let $G$ be an $n$-vertex graph with the maximum degree $Delta$ and the minimum degree $delta$. We give algorithms with complexity $O(1.3158^{n-0.7~Delta(G)})$ and $O(1.32^{n-0.73~Delta(G)})$ that determines if $G$ is 3-colorable, when $delta(G)geq 8$ and $delta(G)geq 7$, respectively.
We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion for compositions, words, matchings, binary trees and permutations, we find generating functions enumerating these objects with respect to their degree of asymmetry, and we describe the limiting distribution of this statistic in each case.
Caccetta-Haggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kuhn and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, i.e., the smaller of the minimum indegree and the minimum outdegree, that forces a large oriented graph to contain a directed cycle of a given length not divisible by $3$, and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by $3$ yet not equal to $3$, hence fully resolve the conjecture of Kelly, Kuhn and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.
Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $sigma(S)geq sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $sigma (K_{r+1}-U, n)$ for $ngeq 5r+18, r+1 geq k geq 7,$ $j geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} bigcup P_{3})$ but not contains a cycle on 4 vertices and not contains $Z_4$. (for example, $C_3bigcup C_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 5)$, $C_3bigcup P_{i_1} bigcup P_{i_2} bigcup ... bigcup P_{i_p}$ $(i_1 geq 3)$, $C_3bigcup P_{i_1} bigcup C_{i_2} bigcup >... bigcup C_{i_p}$ $(i_j eq 4, j=2,3,..., p, i_1 geq 3)$, etc)
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