Contractibility techniques as a closure concept

We introduce a closure concept in the class of line graphs and claw‐free graphs based on contractibility of certain subgraphs in the line graph preimage. The closure can be considered as a common generalization and strengthening of the reduction techniques of Catlin and Veldman and of the closure concept introduced by the first author. We show that the closure is uniquely determined and the closure operation preserves the circumference of the graph. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 37–48, 2003     

Cubic one-regular graphs of order twice a square-free integer

A graph is one-regular if its automorphism group acts regularly on the set of its arcs.Let n be a square-free integer.In this paper,we show that a cubic one-regular graph of order 2n exists if and only if n=3~tp1p2…p_s≥13,where t≤1,s≥1 and p_i's are distinct primes such that 3|(P_i—1). For such an integer n,there are 2~(s-1) non-isomorphic cubic one-regular graphs of order 2n,which are all Cayley graphs on the dihedral group of order 2n.As a result,no cubic one-regular graphs of order 4 times an odd square-free integer exist.     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

Maximizing H‐Colorings of a Regular Graph

For graphs G and H, a homomorphism from G to H, or H‐coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this article is the maximum number of H‐colorings admitted by an n‐vertex, d‐regular graph, for each H. Specifically, writing for the number of H‐colorings admitted by G, we conjecture that for any simple finite graph H (perhaps with loops) and any simple finite n‐vertex, d‐regular, loopless graph G, we have where is the complete bipartite graph with d vertices in each partition class, and is the complete graph on vertices.Results of Zhao confirm this conjecture for some choices of H for which the maximum is achieved by . Here, we exhibit for the first time infinitely many nontrivial triples for which the conjecture is true and for which the maximum is achieved by .We also give sharp estimates for and in terms of some structural parameters of H. This allows us to characterize those H for which is eventually (for all sufficiently large d) larger than and those for which it is eventually smaller, and to show that this dichotomy covers all nontrivial H. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed H, for all d‐regular G, we have where as . More precise results are obtained in some special cases.     

Classification of a family of symmetric graphs with complete 2-arc-transitive quotients

In this paper we give a classification of a family of symmetric graphs with complete 2-arc-transitive quotients. Of particular interest are two subfamilies of graphs which admit an arc-transitive action of a projective linear group. The graphs in these subfamilies can be defined in terms of the cross ratio of certain 4-tuples of elements of a finite projective line, and thus may be called the second type ‘cross ratio graphs’, which are different from the ‘cross ratio graphs’ studied in [A. Gardiner, C. E. Praeger, S. Zhou, Cross-ratio graphs, J. London Math. Soc. (2) 64 (2001), 257–272]. We also give a combinatorial characterisation of such second type cross ratio graphs.     

一类Kn-同胚图色等价类的结构特征

用K（s，n）表示完全图Kn的一条边被长为s（s≥2）的路Ps＋1替代后得到的图．对n≥7，且n-2为素数，刻画了色等价类【K（s，n）]中图的结构特征，进一步，证明了任意任意n≥7，且n-2为素数，K（2，n），K（3，n）是色唯一的．     

On the Dot Product Graph of a Commutative Ring

Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × … ×A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R* = R?{(0, 0,…, 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)* = Z(R)?{(0, 0,…, 0)}. It follows that each edge (path) of the classical zero-divisor graph Γ(R) is an edge (path) of ZD(R). We observe that if n = 1, then TD(R) is a disconnected graph and ZD(R) is identical to the well-known zero-divisor graph of R in the sense of Beck–Anderson–Livingston, and hence it is connected. In this paper, we study both graphs TD(R) and ZD(R). For a commutative ring A and n ≥ 3, we show that TD(R) (ZD(R)) is connected with diameter two (at most three) and with girth three. Among other things, for n ≥ 2, we show that ZD(R) is identical to the zero-divisor graph of R if and only if either n = 2 and A is an integral domain or R is ring-isomorphic to ?₂ × ?₂ × ?₂.     

一类变换图的连通性

本文定义了一类由给定的一个３－正则平面偶图的全体完美匹配所构成的变换图，并证明了该变换图是连通的．由此可得出结论：从任一给定的３－正则平面偶图的完美匹配出发，通过一种所谓的旋转运算，就可以生成全部其它的完美匹配．     

On the fastest moving off from a vertex in directed regular graphs

Let Γ be a directed regular locally finite graph, and let $\bar \Gamma $ be the undirected graph obtained by forgetting the orientation of Γ. Let x be a vertex of Γ and let n be a nonnegative integer. We study the length of the shortest directed path in Γ starting at x and ending outside of the ball of radius n centered at x in $\bar \Gamma $ .     

A class of amply regular graphs related to the subconstituents of a dual polar graph

The connected components of the induced graphs on each subconstituent of the dual polar graph of the odd dimensional orthogonal spaces over a finite field are shown to be amply regular. The connected components of the graphs on the second and third subconstituents are shown to be distance-regular by elementary methods.     

On the action of a group on a graph, II

This paper is a continuation of the survey by the author (V.I. Trofimov, On the action of a group on a graph, Acta Appl. Math. 29 (1992) 161–170) on some results concerning groups of automorphisms of locally finite vertex-symmetric graphs.     

The path minimises the average size of a connected induced subgraph

We prove that among connected graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees, as well as giving a new, shorter proof of the latter.A different proof of the main result was given independently and almost simultaneously by Andrew Vince; the two preprints were submitted one day apart.     

一个伴随等价定理的推广及其简证

记 Gr为任意图 G的 r个拷贝中的对应点 ( r个 )分别与星图 Sr+ 1 的 r个 1度点粘接后得到的图 ,又记 H r为该图 G的相应点与星图 Sr+ 1 的 r度点粘接后得到的图 .如果 G不含三角形 ,则图 ( r- 1) K1 ∪ Gr和图 ( r- 1) G∪ H r伴随等价 ,进而它们的补图色等价     

The color space of a graph

If k is a prime power, and G is a graph with n vertices, then a k‐coloring of G may be considered as a vector in $\mathbb{GF}$(k)ⁿ. We prove that the subspace of $\mathbb{GF}$(3)ⁿ spanned by all 3‐colorings of a planar triangle‐free graph with n vertices has dimension n. In particular, any such graph has at least n − 1 nonequivalent 3‐colorings, and the addition of any edge or any vertex of degree 3 results in a 3‐colorable graph. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 234–245, 2000     

全矩阵环上理想包含图的自同构群

设R是一个环,其上的理想包含图,记为Γ_I(R),是一个有向图,它以R的非平凡左理想为顶点,从R的左理想I_1到I_2有一条有向边当且仅当I_1真包含于I_2.环R上的理想关系图,记为Γ_i(R),也是一个有向图,它以R为顶点集,从R中元素A到B有一条有向边当且仅当A生成的左理想真包含于B生成的左理想.设F_q为有限域,其上n阶全矩阵环记为M_n(F_q),本文刻画了环M_n(F_q)上的理想包含图以及理想关系图的任意自同构.     

关于一类边裂图的k-边幻性

通常没有有效的方法判别一般图G的k-边幻性.本文采用分析方法,讨论了一类非均匀边裂图SPE(Cn,h)的边幻性和k-边幻性,得到一些新的结果.     

Random graphons and a weak Positivstellensatz for graphs

In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2‐variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a “sum of squares” in the algebra of partially labeled graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Circulant Double Coverings of a Circulant Graph of Valency Four

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices. The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com²MaC-KOSEF (R11-1999-054) in Korea.