Characterizing C6＋P2-graphic Sequences

For a given graph H, a graphic sequence π =(d1, d2, ···, dn) is said to be potentially H-graphic if π has a realization containing H as a subgraph. In this paper, we characterize the potentially C6+ P2-graphic sequences where C6+ P2 denotes the graph obtained from C6 by adding two adjacent edges to the three pairwise nonadjacent vertices of C6. Moreover, we use the characterization to determine the value of σ(C6+ P2, n).

The Smallest Degree Sum That Yields Potentially Kr＋1 - K3-Graphic Sequences

Let a（Kr,＋1 - K3,n） be the smallest even integer such that each n-term graphic sequence п= （d1,d2,…dn） with term sum σ（п） = d1 ＋ d2 ＋…＋ dn 〉 σ（Kr＋1 -K3,n） has a realization containing Kr＋1 - K3 as a subgraph, where Kr＋1 -K3 is a graph obtained from a complete graph Kr＋1 by deleting three edges which form a triangle. In this paper, we determine the value σ（Kr＋1 - K3,n） for r ≥ 3 and n ≥ 3r＋ 5.

The smallest degree sum that yields potentially K_(r,r)-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(Kr,r,n) such that every n-term graphic sequence π = (d1,d2,...,dn) with term sum σ(π) = d1 + d2 + ... + dn ≥ σ(Kr,r,n) is potentially Kr,r-graphic, where Kr,r is an r × r complete bipartite graph, i.e. π has a realization G containing Kr,r as its subgraph. In this paper, the values σ(Kr,r,n) for even r and n ≥ 4r2 - r - 6 and for odd r and n ≥ 4r2 + 3r - 8 are determined.

A Characterization for a Sequence to be Potentially K_(r+1) — e-graphic

Let n 〉 r, let lr --- （dl,d2,-,dn） be a non-increasing sequence of nonnegative integers and let Kr＋l - e be the graph obtained from Kr＋l by deleting one edge. If zr has a realization G containing Kr＋l - e as a subgraph, then r is said to be potentially Kr＋l - e-graphic. In this paper, we give a characterization for a sequence π to be potentially Kr＋l - e-graphic.

A Rao-type Characterization for a Sequence to Have a Realization Containing an Arbitrary Subgraph H

Let G be an arbitrary spanning subgraph of the complete graph Kr＋1 on r＋1 vertices and Kr＋1-E（G） be the graph obtained from Kr＋1 by deleting all edges of G.A non-increasing sequence π=（d1,d2,...,dn） of nonnegative integers is said to be potentially Kr＋1-E（G）-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr＋1-E（G） as a subgraph.In this paper,a characterization of π that is potentially Kr＋1-E（G）-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.

Graphic Sequences with a Realization Containing Intersecting Cliques

Let r ≥1, k ≥ 2 and Fm1,… k;r denote the most general definition of a friendship graph,

关于蕴含P_k可图序列的ERDS-JACOBSON-LEHEL猜想

设σ（ｋ，ｎ）是具有下述性质的最小正偶数，每个度和至少为σ（ｋ，ｎ）且没有零项的ｎ项可图序列都是蕴含Ｐｋ可图的．本文给出了当ｋ５，２ｋ＋２ｎ５ｋ－１２时，σ（ｋ，ｎ）的一个下界，并确定了ｋ＝５，６，７时，σ（ｋ，ｎ）的值，即证明了Ｅｒｄｏｓ－Ｊａｃｏｂｓｏｎ－Ｌｅｈｅｌ关于σ（ｋ，ｎ）的猜想对ｋ＝５且ｎ１３，ｋ＝６且ｎ１５，以及ｋ＝７且ｎ１７时成立．

极值图论与度序列

本文简要概述极值图论与度序列的最新研究进展,同时提出了一些有待进一步解决的问题和猜想。

蕴含K5-E(P2)的正可图序列

刻画蕴含K5-E(P2)的正可图序列,其中K5-E(P2)表示从K5中删去两条相邻的边后得到的图。

蕴含W_5可图序列的最小度和

Gould,Jacobson和Lehel考虑了下述经典Tur偄n型极值问题的变形:对于给定的图H,确定最小的正偶数σ(H,n),使得对于每一个n项可图序列π=(d1,d2,…,dn),当σ(π)=d1+d2+…+dn≥σ(H,n)时,π有一个实现G包含H作为可图的.本文确定了当n≥11时,σ(W5,n)之值,其中Wr是r个顶点的轮图.

图的度序列

图的度序列是图论研究中一个重要的课题．至今已发表了４００余篇文章．本文概述这一课题的某些进展，其中包括了可图序列的判准、蕴含Ｐ可图序列和强迫Ｐ可图序列的一些主要结论，同时列出了一些有待进一步研究的问题．

蕴含kC_6的可图序列

刻画了蕴含3C64、C6以及5C6的可图序列,其中一个图G称为具有性质kCl,如果G含有长依次为k,k+1,…,l的圈.

蕴含K_5-Z_4可图序列的刻划

对于给定的图H,称π是蕴含H可图的,如果π有一个实现包含H作为子图.K k,C k,Pk分别表示k阶完全图,圈长为k的圈和路长为k的路.Z 4表示K 4-P2.K 5-Z4表示从5阶完全图中删去Z 4的4条边.本文刻划了当n≥5时,蕴含K 5-Z4的可图序列.

蕴含K_4-e可图序列的刻划

如果1(d=p,2d,,nd)是非增n项可图正整数序列, 4n,则p是蕴含K4-e可图的,当且仅当2,3421ddd且)3(6p,)2,3(42p,)2,3(32p。

蕴含K_4+P_2-可图序列的刻划(英文)

对于给定的图H,若存在可图序列π的一个实现包含H作为子图,则称π为蕴含H-可图的.Gould等人考虑了下述极值问题的变形:确定最小的偶整数σ(H,n),使得每个满足σ(π)≥σ(H,n)的n项可图序列π=(d_1,d_2,…,d_n)是蕴含H-可图的,其中σ(π)=∑d_i.本文刻划了蕴含K_4+P_2-可图序列,其中K_4+P_2是向K4的一个顶点添加两条悬挂边后构成的简单图.这一刻划导出σ(K_4+P_2,n)的值.

论自补图的构造(Ⅰ)

本文通过剖析了4n阶自补图的结构,获得了一些关于自补图的重要性质,并且应用这些结果解决了至今悬而未解的问题—4n阶自补图的构造。

蕴含K_(1,1,3)的正可图序列的最小度和

Gould R J等人考虑了下述经典Turan型极值问题的变形:对于给定的图H,确定最小的正偶数σ(H,n),使得对于每一个n项正可图序列π=(d1,d2,…,dn),当σ(π)=d1+d2+…+dn≥σ(H,n)时,π有一个实现G以H作为子图.本文完全确定了σ(K1,1,3,n)之值,其中Kr,s,t是r×s×t完全三部图.

蕴含K_(1,t)+e的可图序列

本文刻划了蕴含K1,t+e的可图序列,其中t≥3,而K1,t+e是向完全二部图K1,t添加一条边后构成的简单图.

蕴含K_(r+1)-K_(1,t)可图序列的极值问题(英文)

序列S称为蕴含K_m-H可图序列如果S有一个实现包含K_m-H作为子图,本文给出了序列π∈GS_n是蕴含 K_(r+1)- K_(1,t)可图序列的充分条件.

蕴含K_(r,s,t)可图序列的一个极值问题

Gou ld,Jacobson和Lehe l考虑了下述经典Turán型极值问题的一个变形:对于给定的图H,确定最小的偶数σ(H,n),使得每一个n项可图序列π=(d1,d2,…,dn),当σ(π)=d1+d2+…+dn≥σ(H,n)时,π都有一个实现G包含H作为子图.本文确定了σ(K1,2,2,n),8≥n≥5,及当n≥6时,σ(K2,2,2,n)之值,其中Kr,s,t是r×s×t完全三部图.