The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.     

广义字典序积的自同态幺半群

讨论了图的广义字典序积的自同态幺半群的性质,给出了广义字典序积图X[Yz|x∈V（X）]的自同态幺半群与X,Yx(x∈V(X))的自同态幺半群的圈积相等的充要条件。     

Toll number of the Cartesian and the lexicographic product of graphs

Toll convexity is a variation of the so-called interval convexity. A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between $u,v\in V\left(G\right)$ is a set $T_G(u,v)=\{x\in V\left(G\right):x\text{ lies on a tolled walk between }u\text{ and }v\}$. A set $S?V\left(G\right)$ is toll convex, if $T_G(u,v)?S$ for all $u,v\in S$. A toll closure of a set $S?V\left(G\right)$ is the union of toll intervals between all pairs of vertices from $S$. The size of a smallest set $S$ whose toll closure is the whole vertex set is called a toll number of a graph $G$, $tn\left(G\right)$. The first part of the paper reinvestigates the characterization of convex sets in the Cartesian product of two graphs. It is proved that the toll number of the Cartesian product of two graphs equals 2. In the second part, the toll number of the lexicographic product of two graphs is studied. It is shown that if $H$ is not isomorphic to a complete graph, $tn(G°H)\le 3?tn\left(G\right)$. We give some necessary and sufficient conditions for $tn(G°H)=3?tn\left(G\right)$. Moreover, if $G$ has at least two extreme vertices, a complete characterization is given. Furthermore, graphs with $tn(G°H)=2$ are characterized. Finally, the formula for $tn(G°H)$ is given — it is described in terms of the so-called toll-dominating triples or, if $H$ is complete, toll-dominating pairs.     

Mobility of vertex-transitive graphs

We define the mobility of a graph automorphism as the minimum distance between a vertex of the graph and its image under the automorphism, and the absolute mobility of a graph as the maximum of the mobilities of its automorphisms. In this paper, we investigate the mobility of certain classes of graphs, in particular, Cartesian and lexicographic products, vertex-transitive graphs, and Cayley graphs.     

Structure of vertex-transitive graphs

Every vertex-transitive graph has a characteristic structure. The specific details of structure of a vertex-transitive graph are closely related to the configuration of the parts of any minimum separating set of that graph. It is shown for a vertex-transitive graph G that (i) the number n of isomorphic atomic parts admitted by any minimum separating set S is unique for that G; (ii) G is then isomorphic to a disjoint union of m(≥2) copies of such a set of n atomic parts together with some additional edges joining them; (iii) any minimum separating set S of G consists of the vertex set of the union of some k(≥1) copies of the set of n atomic parts; (iv) at most one nonatomic part will be admitted in conjunction with one or more atomic parts by any minimum separating set of G.This configuration of structure extends to nonatomic parts. A vertex-transitive graph G may contain a not necessarily unique minimum separating set S which admits t(≥3) nonatomic parts. Then it is shown that (i) some (t ? 1)-set of these nonatomic parts are pairwise isomorphic; (ii) if the remaining nonatomic part is nonisomorphic to the others then it contains more vertices than the other parts; (iii) G is isomorphic to a disjoint union of d(≥2) copies of the set of q(≥t ? 1) isomorphic nonatomic parts together with some additional edges joining them; (iv) a minimum separating set S consists of the vertex set of the union of some y(≥1) copies of the set of q isomorphic nonatomic parts. For the case of t = 2 non-atomic parts admitted by a minimum separating set S, counterexamples of (iii) and (iv) are given.     

Classifying vertex-transitive graphs whose order is a product of two primes

Vertex-transitive graphs whose order is a product of two primes with a primitive automorphism group containing no imprimitive subgroup are classified. Combined with the results of [15] a complete classification of all vertex-transitive graphs whose order is a product of two primes is thus obtained (Theorem 2.1).Supported in part by the Research Council of SloveniaSupported in part by the Italian Ministry of Research (MURST)     

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs. We characterize these graphs by finite combinatorial objects called labeling schemes. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of given degree, as well as most of their transitive groups of automorphisms. In addition,we are able to decide whether a given TLF-planar transitive graph is Cayley or not. This class contains all the one-ended planar Cayley graphs and the normal transitive tilings of the plane.     

On self-complementary vertex-transitive graphs

In this paper we study the problem of determining positive integers n for which there exist self-complementary vertex-transitive graphs of order n. The main aim of this paper is to prove that, for distinct primes p, q, there exist self-complementary vertex-transitive graphs of order pq if and only if p, q Scz.tbnd;1 (mod 4). This provides negative answers to a long-standing open question proposed by Zelinka (1979) and a recent open question proposed by Froncek, Rosa Siran (1996).     

A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G[G], then the wreath product of by itself is a proper subgroup of . A positive answer is provided by constructing an automorphism Ψ of G[G] which satisfies: for every vertex x of G, there is an infinite subset I(x) of V(G) such that Ψ({x}×V(G))=I(x)×V(G).     

Distance degrees of vertex-transitive graphs

In [4], a lower bound of distance degrees of distance degree regular graphs is obtained. In this paper, we prove that a lower bound will be improved in some cases of vertex-transitive graphs.     

Strong cliques in vertex-transitive graphs

A clique (resp, independent set) in a graph is strong if it intersects every maximal independent set (resp, every maximal clique). A graph is clique intersect stable set (CIS) if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique $C$ in a vertex-transitive graph $\mathrm{\Gamma }$ is strong if and only if $\text{◂=▸}\text{◂⋅▸}\mid C\mid \mid I\mid =\mid V\left(\mathrm{\Gamma }\right)\mid$ for every maximal independent set $I$ of $\mathrm{\Gamma }$. On the basis of this result we prove that a vertex-transitive graph is CIS if and only if it admits a strong clique and a strong independent set. We classify all vertex-transitive graphs of valency at most 4 admitting a strong clique, and give a partial characterization of 5-valent vertex-transitive graphs admitting a strong clique. Our results imply that every vertex-transitive graph of valency at most 5 that admits a strong clique is localizable. We answer an open question by providing an example of a vertex-transitive CIS graph which is not localizable.     

Pentavalent vertex-transitive diameter two graphs

We classify the family of pentavalent vertex-transitive graphs Γ with diameter 2. Suppose that the automorphism group of Γ is transitive on the set of ordered distance 2 vertex pairs. Then we show that either Γ is distance-transitive or Γ is one of \(\overline {{C_8}} ,{\kern 1pt} {K_5}\square {K_2},{\kern 1pt} {C_5}\left[ {{K_2}} \right],{\kern 1pt} \overline {2{C_4}} ,{\kern 1pt} or{\kern 1pt} {K_3}\square {K_4}\).     

On the structure of infinite vertex-transitive graphs

A characterization of all vertex-transitive graphs Γ of connectivity l is given in terms of the lobe structure of Γ. Among these, all graphs are determined whose automorphism groups act primitively (respectively, regularly) on the vertex set.     

Vertex-transitive graphs and vertex-transitive maps

We consider vertex-transitive graphs embeddable on a fixed surface. We prove that all but a finite number of them admit embeddings as vertex-transitive maps on surfaces of nonnegative Euler characteristic (sphere, projective plane, torus, or Klein bottle). It follows that with the exception of the cycles and a finite number of additional graphs, they are factor graphs of semiregular plane tilings. The results generalize previous work on the genus of minimal Cayley graphs by V. Proulx and T. W. Tucker and were obtained independently by C. Thomassen, with significant differences in the methods used. Our method is based on an excursion into the infinite. The local structure of our finite graphs is studied via a pointwise limit construction, and the infinite vertex-transitive graphs obtained as such limits are classified by their connectivity and the number of ends. In two appendices, we derive a combinatorial version of Hurwitz's Theorem, and classify the vertex-transitive maps on the Klein bottle.     

Long cycles in vertex-transitive graphs

We prove that every connected vertex-transitive graph on n ≥ 4 vertices has a cycle longer than (3n)^1/2. The correct order of magnitude of the longest cycle seems to be a very hard question.     

关于n-可扩图的一些结果

本文讨论了关于m-可扩图的两个极值问题；并考查了下述图类的n-可扩性；正则偶图，单位区间图和分裂图。