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.     

STABILITY NUMBER IN SUBCLASSES OF P_5~-FREE GRAPHS

§1　IntroductionLet Pn,Cn,and Kn be the path,the cycle,and the complete graph of order n,respectively.The complete bipartite graph with cardinalities ofparts m and n is denoted byKm,n.Disjoint union of n copies of a graph G is denoted by n G;Gis the complement of agraph G.A setof pairwise non-adjacent vertices in a graph is called stable.A maximum stableset of a graph G has the largest cardinalityα( G) among all stable sets of G;α( G) is calledthe stability number of G.Let Z be a s…     

二部图的E-H-不可收缩性

By End（G） and hEnd（G） we denote the set of endomorphisms and half-strong endomorphisms of a graph G respectively. A graph G is said to be E-H-unretractive if End（G） = hEnd（G）. A general characterization of an E-H-unretractive graph seems to be difficult. In this paper, bipartite graphs with E-H-unretractivity are characterized explicitly.     

Minimum congestion spanning trees in bipartite and random graphs

The first problem considered in this article reads: is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n 3 2 , where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n 3 2 .     

（g,f）-FACTORS WITH SPECIAL PROPERTIES IN BIPARTITE （mg,mf）-GRAPHS

Let G be a bipartite graph and g and f be two positive integer-valued functions defined on vertex set V(G) of G such that g(x)≤f(x). In this paper,some sufficient conditions related to the connectivity and edge-connectivity for a bipartite (mg,mf)-graph to have a (g,f)-factor with special properties are obtained and some previous results are generalized.Furthermore,the new results are proved to be the best possible.     

符号混合控制问题的某些NP-完全性结果

Let G =(V, E) be a simple graph with vertex set V and edge set E. A signed mixed dominating function of G is a function f:V∪E→ {-1, 1} such that ∑_(y∈N_m(x)∪{x})f(y)≥ 1for every element x∈V∪E, where N_m(x) is the set of elements of V∪E adjacent or incident to x. The weight of f is w(f) =∑_(x∈V∪E)f(x). The signed mixed domination problem is to find a minimum-weight signed mixed dominating function of a graph. In this paper we study the computational complexity of signed mixed domination problem. We prove that the signed mixed domination problem is NP-complete for bipartite graphs, chordal graphs, even for planar bipartite graphs.     

Total Chromatic Number of the Join of K_(m,n) and C_n

The total chromatic number XT(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if XT(G)=Δ(G) 1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.     

图的循环带宽和

Abstract. Let G be a simple graph. The cyclic bandwidth sum problem is to determine a labeling of graph G in a cycle such that the total length of edges is as small as possible. In this paper, some upper and lower bounds on cyclic bandwidth sum of graphs are studied.     

Extremal Problems on Components and Loops in Graphs

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that Ω(G) ≤-4 are shown to be disconnected, and if Ω(G) ≥-2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G) =-2, then certainly the graph should be a tree.Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤-4, the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs.We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.     

偶匹配可扩性的极图问题(英文)

设G是含有完美匹配的简单图.称图G是偶匹配可扩的(BM-可扩的),如果G的每一个导出子图是偶图的匹配M都可以扩充为一个完美匹配.极图问题是图论的核心问题之一.本文将刻画极大偶匹配不可扩图,偶图图类和完全多部图图类中的极大偶匹配可扩图.     

导出匹配问题的NP-完全性以及导出匹配可扩问题的CO-NP-完全性

图Ｇ的一个匹配Ｍ是导出的,若Ｍ是图Ｇ的一个导出子图。图Ｇ是导邮匹配可扩的（简记ＩＭ－可扩的）,若图Ｇ的任一导出匹配均含于图Ｇ的一个完美匹配当中。本文我们将证明如下结果。⑴对无爪图而言,问题“给定图Ｇ以及一个正整数ｒ,确定是否存在图Ｇ的一个导出匹配Ｍ使得Ｍ≥ｒ”是ＮＰ－完全的。⑵对直径为２的图以及直径为３的偶图,问题“确定一个给定图是否为导出匹配可扩的”是ＣＯ－ＮＰ完全的;而对完全多部图而言,问题“     

A New Graph Structure of Commutative Semigroups

In this paper, a new zero-divisor graph $\overline{\G}(S)$ is defined and studied for a commutative semigroup $S$ with zero element. The properties and the structure of the graph are studied; for any complete graph and complete bipartite graph $G$, commutative semigroups $S$ are constructed such that the graph $G$ is isomorphic to $\overline{\G}(S)$.     

Graham’s pebbling conjecture on product of complete bipartite graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs G and H, f( G x H) ⩽ f( G) f( H). We show that Graham’s conjecture holds true of a complete bipartite graph by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are complete bipartite graphs.     

退化图中的子图计数问题研究

令G表示n个顶点的图,如果G的每个子图中都包含一个度至多为k的顶点,则称G为k-退化图.令N(G,F)表示G中F子图的个数.主要研究了k-退化图中完全子图和完全二部子图的计数问题,给出了计数的上界以及相应的极图.首先,证明了Ν(G,K_t)≤(n-k)(k t-1)+(k t).其次,如果s,t≥1,n≥k+1且s+t≤k,我们证明了Ν(G,K_s,t)≤{(k s)(n-s s)-1/2(k s)(k-s s),t=s,(k s)(n-s t)+(k t)(n-t s)-(k t)(k-t s),t≠s.此外,还研究了在最大匹配和最小点覆盖为给定值的情况下,图G中的最大边数.记v(G),K(G)分别为图G的最大匹配数和最小点覆盖.证明了当v(G)≤k,K(G)=k+r且n≥2k+2r²+r+1时,有e(G)≤(k+r+1 2)+(k-r)(n-k-r-1).     

INVERSE MONOIDS OF GRAPHS

. IntroductionGraph endomorphism and its regularity property have been investigated in some literatures (of. [1--41 for examples). The invertibility is a stronger algebraic property thanregUlarity in semigroup theory. It is commonly agreed that inverse semigroups are the mostpromising class of semigroups for study. In this paper we first present a combinatorial characterization of an inverse monoid of a graph (Theorem 2.3). Then using this we prove thata bipartite graph with an inverse monoi…     

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

The bipartite margin shop and maximum red matchings free of blue–red alternating cycles

The margin shop arises as a model of margining investment portfolios in a batch, a mandatory end-of-day risk management operation for any prime brokerage firm. The margin-shop scheduling problem is the extension of the preemptive flow-shop scheduling problem where precedence constraints can be introduced between preempted parts of jobs. This paper is devoted to the bipartite case which is equivalent to the problem of finding a maximum red matching that is free of blue–red alternating cycles in a complete bipartite graph with blue and red edges. It is also equivalent to the version of the jump-number problem for bipartite posets where jumps inside only one part should be counted. We show that the unit-time bipartite margin-shop scheduling problem is NP-hard but can be solved in polynomial time if the precedence graph is of degree at most two or a forest.     

图的{P4}——分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上．记Pl长度为l-1的路，如果G能够分解成若干个Pl，则称G存在{Pl}——分解，关于图的给定长路分解问题主要结果有：（i）连通图G存在{P3}-分解当且仅当G有偶数条边（见[1]）；（ii）连通图G存在{P3，P4}-分解当且仅当G不是C3和奇树，这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树（见[3]）．本文讨论了3正则图的{P4}--分解情况，并构造证明了边数为3k（k∈Z且k≥2）的完全图Kn和完全二部图Kr,s存在{P4}-分解.