Lili Hu, Chunhui Lai, Ping Wang
Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_4$ and $K_5-Y_4$-graphic sequences where $Y_4$ is a tree on 5 vertices and 3 leaves.
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For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. In this paper, we characterize the potentially $K_6-C_5$ -graphic sequences.
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic sequences where $H$ denotes 3-regular graph with 6 vertices. In other words, we characterize the potentially $K_{3,3}$ and $K_6-C_6$-graphic sequences where $K_{r,r}$ is an $r\times r$ complete bipartite graph. One of these characterizations implies a theorem due to Yin [25].
In this paper, we characterize the potentially $(K_5-C_4)$-graphic sequences where $K_5-C_4$ is the graph obtained from $K_5$ by removing four edges of a 4 cycle $C_4$. This characterization implies a theorem due to Lai [6].
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_6-C_4$-graphic sequences. This characterization implies a theorem due to Hu and Lai [7].
For given a graph $H$, a graphic sequence $π=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $π$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ where $H$ is a subgraph of $K_m$. In this paper, we characterize potentially $K_{2,5}$-graphic sequences. This characterization implies a special case of a theorem due to Yin et al. [26].
Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ of $H$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_5-P_3$, $K_5-A_3$, $K_5-K_3$ and $K_5-K_{1,3}$-graphic sequences where $A_3$ is $P_2\cup K_2$. Moreover, we also characterize the potentially $K_5-2K_2$-graphic sequences where $pK_2$ is the matching consisted of $p$ edges.
A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine $σ(K_{r+1}-(kP_2\bigcup tK_2), n)$ for $n\geq 4r+10, r+1 \geq 3k+2t, k+t \geq 2,k \geq 1, t \geq 0$ .
A sequence $S$ is potentially $K_4-e$ graphical if it has a realization containing a $K_4-e$ as a subgraph. Let $σ(K_4-e, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_4-e, n)$ is potentially $K_4-e$ graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $σ(K_4-e, n)$. In this paper, we prove that $σ(K_4-e, n)=2[(3n-1)/2]$ for $n\geq 7$, and $n=4,5,$ and $σ(K_4-e, 6)= 20$.
A sequence $S$ is potentially $K_{m}-C_{4}$-graphical if it has a realization containing a $K_{m}-C_{4}$ as a subgraph. Let $σ(K_{m}-C_{4}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-C_{4}, n)$ is potentially $K_{m}-C_{4}$-graphical. In this paper, we prove that $σ(K_{m}-C_{4}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq 4.$ We conjecture that equality holds for $n \geq m \geq 4.$ We prove that this conjecture is true for $m=5$.
A sequence $S$ is potentially $K_{m}-P_{k}$ graphical if it has a realization containing a $K_{m}-P_{k}$ as a subgraph. Let $σ(K_{m}-P_{k}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-P_{k}, n)$ is potentially $K_{m}-P_{k}$ graphical. In this paper, we prove that $σ(K_{m}-P_{k}, n)\geq (2m-6)n-(m-3)(m-2)+2,$ for $n \geq m \geq k+1\geq 4.$ We conjecture that equality holds for $n \geq m \geq k+1\geq 4.$ We prove that this conjecture is true for $m=k+1=5$ and $m=k+2=5$.
A sequence of nonnegative integers $π$ is {\it graphic} if it is the degree sequence of some graph $G$. In this case we say that $G$ is a \textit{realization} of $π$, and we write $π=π(G)$. A graphic sequence $π$ is {\it potentially $H$-graphic} if there is a realization of $π$ that contains $H$ as a subgraph. Given nonincreasing graphic sequences $π_1=(d_1,\ldots,d_n)$ and $π_2 = (s_1,\ldots,s_n)$, we say that $π_1$ {\it majorizes} $π_2$ if $d_i \geq s_i$ for all $i$, $1 \leq i \leq n$. In 1970, Erdős showed that for any $K_{r+1}$-free graph $H$, there exists an $r$-partite graph $G$ such tha…
A sequence $S$ is potentially $K_{p,1,1}$ graphical if it has a realization containing a $K_{p,1,1}$ as a subgraph, where $K_{p,1,1}$ is a complete 3-partite graph with partition sizes $p,1,1$. Let $σ(K_{p,1,1}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{p,1,1}, n)$ is potentially $K_{p,1,1}$ graphical. In this paper, we prove that $σ(K_{p,1,1}, n)\geq 2[((p+1)(n-1)+2)/2]$ for $n \geq p+2.$ We conjecture that equality holds for $n \geq 2p+4.$ We prove that this conjecture is true for $p=3$.
The set of all non-increasing nonnegative integers sequence $π=$ ($d(v_1),$ $d(v_2),$ $...,$ $d(v_n)$) is denoted by $NS_n$. A sequence $π\in NS_n$ 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 $π$. The set of all graphic sequences in $NS_n$ is denoted by $GS_n$. A graphical sequence $π$ is potentially $H$-graphical if there is a realization of $π$ containing $H$ as a subgraph, while $π$ is forcibly $H$-graphical if every realization of $π$ contains $H$ as a subgraph. Let $K_k$ denote a complete graph o…
A non-increasing sequence $π=(d_1,\ldots,d_n)$ of nonnegative integers is said to be graphic if it is realizable by a simple graph $G$ on $n$ vertices. A graphic sequence $π=(d_1,\ldots,d_n)$ is said to be potentially $_3C_\ell$-graphic if there is a realization of $π$ containing cycles of every length $r$, $3\le r\le \ell$. It is well-known that if the non-increasing degree sequence $(d_1,\ldots,d_\ell)$ of a graph $G$ on $\ell$ vertices satisfies the Pósa condition that $d_{\ell+1-i}\ge i+1$ for every $i$ with $1\le i<\frac{\ell}{2}$, then $G$ is either pancyclic or bipartite. In this paper,…
A sequence $S$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical if it has a realization containing a $K_{p_{1},p_{2},...,p_{t}}$ as a subgraph, where $K_{p_{1},p_{2},...,p_{t}}$ is a complete t-partite graph with partition sizes $p_{1},p_{2},...,p_{t} (p_{1}\geq p_{2}\geq ...\geq p_{t} \geq 1)$. Let $σ(K_{p_{1},p_{2},...,p_{t}}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{p_{1},p_{2},...,p_{t}}, n)$ is potentially $K_{p_{1},p_{2},...,p_{t}}$ graphical. In this paper, we prove that $σ(K_{p_{1},p_{2},...,p_{t}}, n)\geq 2[((2p_{1}+2p_…
In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $σ(S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$-term graphical sequence $S$ having $σ(S)\ge l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $σ(H, n)$. We show $σ(C_{2m+1}, n)=m(2n-m-1)+2$, for $m\ge 3$, $n\ge 3m$; $σ(C_{2m+2}, n)=m(2n-m-1)+4$, for $m\ge 3, n\ge 5m-2$.
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 $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $σ(K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices a…
A sequence of nonnegative integers π=(d_1,d_2,...,d_n) is graphic if there is a (simple) graph G with degree sequence π. In this case, G is said to realize or be a realization of π. Degree sequence results in the literature generally fall into two classes: forcible problems, in which all realizations of a graphic sequence must have a given property, and potential problems, in which at least one realization of πmust have the given property. Given a graph H, a graphic sequence πis potentially H-graphic if there is some realization of πthat contains H as a subgraph. In 1991, Erdős, Jacobson and L…
For an integer sequence (with even sum), the closer that the sequence is to being regular, the more likely that the sequence is graphic. But how regular must a sequence be before it must always be graphic? We show that for many sequences if all values are within $\frac{n-2}{4}$ of the mean degree value, then the sequence is graphic. We also see how this result extends to show when a maximum difference between sequence values implies that a sequence is graphic.
Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. 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 $K_{m}$). A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we d…
As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, at the expense of possibly producing an erroneous output on some "yes" instances but with very low error rate.
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 $σ(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $σ(S)\geq σ(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $σ(K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices a…
We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.
A Sufficient Condition for Graphic Sequences with Given Largest and Smallest Entries, Length, and Sum
本文给出一个基于度序列最大最小项、长度与和的充分条件，确保其为图序列，并推广了Zverovich与Zverovich的已有结果。
Let $A_n=(a_1,a_2,\ldots,a_n)$ and $B_n=(b_1,b_2,\ldots,b_n)$ be nonnegative integer sequences with $A_n\le B_n$. The purpose of this note is to give a good characterization such that every integer sequence $π=(d_1,d_2,\ldots d_n)$ with even sum and $A_n\le π\le B_n$ is graphic. This solves a forcible version of problem posed by Niessen and generalizes the Erdős--Gallai theorem.
Forcibly unicyclic and bicyclic graphic sequences
本文研究图序列的强性质，旨在刻画强单圈和强双圈图序列，即所有具有该度序列的图均为连通且边数分别为 n 或 n+1 的图。
Let $π_1=(d_1^{(1)}, \ldots,d_n^{(1)})$ and $π_2=(d_1^{(2)},\ldots,d_n^{(2)})$ be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $π_1$ and $π_2$, respectively, on vertex set $\{v_1,\dots,v_n\}$ such that for $j\in\{1,2\}$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $i\in\{1,\ldots,n\}$. In this case, we say that $(G_1,G_2)$ is a $(π_1,π_2)$-\textit{packing}. A clear necessary condition for graphic sequences $π_1$ and $π_2$ to pack is that $π_1+π_2$, their componentwise sum, is also graphic. It is known, however, that this condition is not sufficie…
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.
A sequence of nonnegative integers $π=(d_1,d_2,...,d_n)$ is graphic if there is a (simple) graph $G$ of order $n$ having degree sequence $π$. In this case, $G$ is said to realize or be a realization of $π$. Given a graph $H$, a graphic sequence $π$ is potentially $H$-graphic if there is some realization of $π$ that contains $H$ as a subgraph. In this paper, we consider a degree sequence analogue to classical graph Ramsey numbers. For graphs $H_1$ and $H_2$, the potential-Ramsey number $r_{pot}(H_1,H_2)$ is the minimum integer $N$ such that for any $N$-term graphic sequence $π$, either $π$ is p…
Any fully graphic region of degree sequences can be sampled rapidly
本文研究度序列区域的稳定性与采样问题，证明全可图度序列区域满足$3n^{13}$-稳定性、对应switch马尔可夫链可快速混合，同时解决了此前提出的反向相关问题。
There are a variety of existing conditions for a degree sequence to be graphic. When a degree sequence satisfies any of these conditions, there exists a graph that realizes the sequence. We formulate several novel sufficient graphicality criteria that depend on the number of elements in the sequence, corresponding to the number of nodes in an associated graph, and the mean degree of the sequence. These conditions, which are stated in terms of bidegree sequences for directed graphs, are easier to apply than classic necessary and sufficient graphicality conditions involving multiple inequalities…
Small-scale operations on graphic sequences
本文引入2-约化操作，通过从两个整数中各减1保持图性，用于生成所有实现、证明现有刻画并提出新刻画，突出图序列的数学与算法联系。
On generating $k$-factorable graphic sequences with connected (resp.no connected) $k$-factors
本文研究如何生成带有连通或不连通$k$-因子的可$k$-因子化图序列这一构造问题。
We present infinite sequences of excluded minors for both the class of lifted-graphic matroids and the class of frame matroids.
On triangular biregular degree sequences
本文研究了三角图与度序列问题，猜想所有项均大于等于4的图度序列都有三角实现，并对一类双正则图度序列证明了该猜想。
A graphic sequence $π$ is potentially $H$-graphic if there is some realization of $π$ that contains $H$ as a subgraph. The Erdős-Jacobson-Lehel problem asks to determine $σ(H,n)$, the minimum even integer such that any $n$-term graphic sequence $π$ with sum at least $σ(H,n)$ is potentially $H$-graphic. The parameter $σ(H,n)$ is known as the potential function of $H$, and can be viewed as a degree sequence variant of the classical extremal function ${\rm ex}(n,H)$. Recently, Ferrara, LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree sequence is potentially $H$-graphic…
An Efficient Algorithm to Generate all Labeled Triangle-free Graphs with a given Graphical Degree Sequence
本文提出了一种高效算法，用于生成具有特定图度序列的无三角形标号图，并通过剪枝技术提高效率。
For finite sequence $\underbar{\em d}$ of positive integers, we consider graphs that have $\underbar{\em d}$ as their list of vertex degrees, and bipartite graphs for which each part has $\underbar{\em d}$ as its list of vertex degrees. In particular, we make a connection between a result for bipartite graphs by Alon, Ben-Shimon and Krivelevich and a result of Zverovich and Zverovich for graphs, and we give an improvement of a result of Zverovich and Zverovich. We show that the bipartite graphs with vertex degree sequences $(\underbar{\em d},\underbar{\em d}\,)$ are in one to one correspondenc…
Let $n$ be even, let $π= (d_1, \ldots, d_n)$ be a graphic degree sequence, and let $π- k = (d_1 - k, \ldots, d_n - k)$ also be graphic. Kundu proved that $π$ has a realization $G$ containing a $k$-factor, or $k$-regular graph. Another way to state the conclusion of Kundu's theorem is that $π$ \emph{potentially} contains a $k$-factor. Busch, Ferrara, Hartke, Jacobsen, Kaul, and West conjectured that more was true: $π$ potentially contains $k$ edge-disjoint $1$-factors. Along these lines, they proved $π$ would potentially contain edge-disjoint copies of a $(k-2)$-factor and two $1$-factors. We f…
The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular graphic bicyclic sequences. Moreover, we obtain the relations of the second Zagreb indices among the extremal graphs with different degree sequences.
With the current burst of network theory (especially in connection with social and biological networks) there is a renewed interest on realizations of given degree sequences. In this paper we propose an essentially new degree sequence problem: we want to find graphical realizations of a given degree sequence on labeled vertices, where certain would-be edges are {\em forbidden}. Then we want to sample uniformly and efficiently all these possible realizations. (This problem can be considered as a special case of Tutte's $f$-factor problem, however it has a favorable sampling speed.) We solve thi…
A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$ is an 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $π=(d_1,\ldots,d_n)$ of nonnegative integers is a {\it graphic sequence} if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795--802) proved that if $k\ge 2$, $n\ge \frac{9}{2}k^2+\frac{19}{2}k$ and $π=(d_1,\ldots,d_n)$ is a graphic sequence with $\sum\limits_{i=1}^n d_i>(k-2)n$, then $π$ has a re…
Fully graphic degree sequences and P-stable degree sequences
本文研究图度序列的P-稳定性与完全图性，证明已知P-稳定集合均为完全图，并在Σ不出现时证明两者等价，加强Jerrum-McKay-Sinclair定理，并提出Σ出现时等价性的猜想。
Degree sequences of triangular multigraphs
本文研究三角多重图的度序列问题，将Egan和Nikolayevsky关于三角简单图的猜想推广到多重图情形，并证明了每一项至少为4的可图序列均可实现为三角多重图的度序列。
One of the first graph theoretical problems which got serious attention (already in the fifties of the last century) was to decide whether a given integer sequence is equal to the degree sequence of a simple graph (or it is {\em graphical} for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the {\em swap} operation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another one of the same degree sequence. This latter problem got particular emphases in connection of fast …
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.
S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been settled completely by M. Chudnovsky and P. Seymour. One part of the proof proves the result for the bounded case, a result proved independently by C. J. Altomare. We give two short proofs of the bounded case of S. B. Rao's conjecture. Both the proofs use the fact that if the number of entries in an integer sequence (with even sum) is much larger than its highest term, then it is necessarily graphic.
We give characterizations of the structure and degree sequences of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs. The degree sequence characterization of hereditary unigraphs makes use of the list of values $k$ for which the $k$th Erdős--Gallai inequality holds with equality for a graphic sequence. Using the canonical decompositi…
S.B. Rao conjectured in 1971 that graphic degree sequences are well quasi ordered by a relation defined in terms of the induced subgraph relation. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao's Conjecture by giving structure theorems for graphic degree sequences. In this paper, we prove and use a variant of Dickson's Lemma from commutative semigroup theory to give a short proof of the bounded degree case of Rao's Conjecture that is independent of the Chudnovsky-Seymour structure theory. In fact, we affirmatively answer two questions of N. Robertson, the first of which im…
The log concavity of two graphical sequences
本文证明了任意图的大笛卡尔幂关于固定顶点的价序列及距离正则图的价序列均具有对数凹性，改进了已有结果，并推导出强正则图、二权码、完全正则码及Q-多项式结合方案重数序列的相关结论。
In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\m…
The sequence of prime gaps is graphic
本文提出素数间隙图的概念，证明在黎曼猜想假设下对任意 $n\geq 2$ 均存在，并可通过保度增长过程生成无限序列，首次揭示了自然数无限序列的图示性。
Common neighbours in planar graphs
本文研究平面图中不同顶点共同邻居数量的分类问题，提出了一种完整分类方法，解决了平面图 $n$-度序列问题。
Separator-based derivations of graphic arrangements
本文针对非自由情况下的图形超平面安排，基于图分隔子引入了对数导子模的显式生成元集，获得了导子度序列的新见解并给出了最高度界限。
Assume that we are given two graphic sequences, $π_1$ and $π_2$. We consider conditions for $π_1$ and $π_2$ which guarantee that there exists a simple graph $G_2$ realizing $π_2$ such that $G_2$ is the subgraph of any simple graph $G_1$ that realizes $π_1$.
Counting graphic sequences via integrated random walks
本文通过积分随机游走方法，将图序列计数函数$G(n)$的上下界改进了$n^{1/4}$倍，扩展了精确值范围，并确定了随机游走桥积分非负的渐近概率。
New results on graph matching from degree preserving growth
本文研究 Degree Preserving Growth 模型中的匹配问题，证明度 2δ 扩展可图性等价于存在大小 δ 的匹配，并给出强制与潜在匹配数的界。
Dense, irregular, yet always graphic $3$-uniform hypergraph degree sequences
本文针对3-均匀超图度序列问题，证明了当所有度落在$\ rac{2n^2}{63}+O(n)$与$\ rac{5n^2}{63}-O(n)$之间、顶点数至少45且度和可被3整除时，度序列必可实现，并首次给出构造高度不规则稠密度序列的多项式时间算法。
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have multiple inputs and outputs. If such a construction is possible, then determining whether the studied semigroup is symmetric, or finding explicit forms of its Apéry set and Hilbert series, is reduced to straightforward computations assisted by a MAPLE program we made available on arXiv. This approach applies to many of the cases considered in literature, inc…
S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been proved by Chudnovsky and Seymour. We give an independent short proof of the labelled version of the almost bounded case of S. B. Rao's conjecture, the case where we have a bound on the degree, but allow a bounded number of vertices to have unbounded degree.
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