Mark Watkins
Syracuse University, Mathematics, Emeritus
In this note, G will denote a finite undirected graph without multiple edges, and V = V(G) will denote its vertex set. The largest integer n for which G is n-vertex connected is the vertex-connectivity of G and will be denoted by λ =... more
In this note, G will denote a finite undirected graph without multiple edges, and V = V(G) will denote its vertex set. The largest integer n for which G is n-vertex connected is the vertex-connectivity of G and will be denoted by λ = λ(G). One defines ζ to be the largest integer z not exceeding |V| such that for any set U ⊂ V with |U| = z, there is a cycle in G which contains U. The symbol i(U) will denote the component index of U. As a standard reference for this and other terminology, the authors recommend O. Ore (3).
Research Interests:
A tessellation of the plane is face-homogeneous if for some integer k≥3 there exists a cyclic sequence σ=[p_0,p_1,...,p_k-1] of integers ≥3 such that, for every face f of the tessellation, the valences of the vertices incident with f are... more
A tessellation of the plane is face-homogeneous if for some integer k≥3 there exists a cyclic sequence σ=[p_0,p_1,...,p_k-1] of integers ≥3 such that, for every face f of the tessellation, the valences of the vertices incident with f are given by the terms of σ in either clockwise or counter-clockwise order. When a given cyclic sequence σ is realizable in this way, it may determine a unique tessellation (up to isomorphism), in which case σ is called monomorphic, or it may be the valence sequence of two or more non-isomorphic tessellations (polymorphic). A tessellation which whose faces are uniformly bounded in the Euclidean plane is called a Euclidean tessellation; a non-Euclidean tessellation whose faces are uniformly bounded in the hyperbolic plane is called hyperbolic. Hyperbolic tessellations are well-known to have exponential growth. We seek the face-homogeneous hyperbolic tessellation(s) of slowest growth and show that the least growth rate of monomorphic face-homogeneous tess...
Research Interests:
A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of... more
A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of an element a in A is the number of points of X that are moved by a, and the motion of the group A is the minimal motion of its nonidentity elements. For finite A, the Motion Lemma says that if the motion of A is large enough (specifically at least 2 log_2 |A|), then the action is 2-distinguishable. For many situations where X has a combinatorial or algebraic structure, the Motion Lemma implies the action of Aut(X) on X is 2-distinguishable in all but finitely many instances. We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee 2-distinguishability. From this we deduce a number of results, including the fact that every locally finite, connected gr...
Research Interests:
A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k parts such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that... more
A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k parts such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G,V) is k-distinguishable is its distinguishing number. In particular, a graph X is k-distinguishable if its automorphism group Aut(X) has distinguishing number at most k in its action on the vertices of X. Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph X is not k-distinguishable (for a given cardinal k), then it contains a ball B of finite radius whose distinguishing number is at least k. Moreover, this lower bound cannot be sharpened, since for any integer k greater than 3 there exists an infinite, locally finite, connected graph X that is not k-distinguishable but in which e...
Research Interests:
A Bilinski diagram (respectively, B∗-diagram) is a labeling of a planar map with respect to the regional distance of its vertices and faces from a central vertex (respectively, face). Such diagrams are concentric if for each k ≥ 1, the... more
A Bilinski diagram (respectively, B∗-diagram) is a labeling of a planar map with respect to the regional distance of its vertices and faces from a central vertex (respectively, face). Such diagrams are concentric if for each k ≥ 1, the set of vertices at regional distance k from the central vertex or face induces a circuit. The class Ga,b consists of all 1-ended, 3-connected planar maps with the property that every valence is finite and at least a and every covalence is finite and at least b. A map in the subclass Ga,b+ of Ga,b contains no adjacent b-covalent faces, and dually a map in Ga+,b contains no adjacent a-valent vertices. It is shown that all Bilinski diagrams and all B∗-diagrams of all maps in G6,3, G4,4, G3,6, G5,3+ and G3+,5 are concentric.
The distinguishing number of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The distinguishing number of a... more
The distinguishing number of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have connectivity 1 if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_\alpha$ is called a suborbit of $G$.We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number $2$. Consequently, any nonnull, infinite, primitive, locally finite graph is $2$-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denum...
Research Interests:
A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number... more
A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is no...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph... more
The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph Γ is said to have connectivity 1 if there exists a vertex α∈ VΓ such that Γ∖{α} is not connected. For α∈ V, an orbit of the point stabilizer G_α is called a suborbit of G. We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our re...
Research Interests:
Research Interests:
The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse... more
The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.
Research Interests:
A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts... more
A graphical Frobenius representation (GFR) of a Frobenius (permutation) group $G$ is a graph $\Gamma$ whose automorphism group Aut$(\Gamma)$ acts as a Frobenius permutation group on the vertex set of $\Gamma$, that is, Aut$(\Gamma)$ acts vertex-transitively with the property that all nonidentity automorphisms fix either exactly one or zero vertices and there are some of each kind. The set $K$ of all fixed-point-free automorphisms together with the identity is called the kernel of $G$. Whenever $G$ is finite, $K$ is a regular normal subgroup of $G$ (F. G. Frobenius, 1901), in which case $\Gamma$ is a Cayley graph of $K$. The same holds true for all the infinite instances presented here.Infinite, locally finite, vertex-transitive graphs can be classified with respect to (i) the cardinality of their set of ends and (ii) their growth rate. We construct families of infinite GFRs for all possible combinations of these two properties. There exist infinitely many GFRs with polynomial growth...
Research Interests:
Research Interests:
Research Interests:
Without Abstract
Research Interests:
... In Section 3 we present our main constructive tool; we prove that the derived graphs of nearly all vertex-and edge-transitive, odd-valent, non-bipartite graphs are VTNCG's. In Section 4 we construct finite VTNCG's, some with... more
... In Section 3 we present our main constructive tool; we prove that the derived graphs of nearly all vertex-and edge-transitive, odd-valent, non-bipartite graphs are VTNCG's. In Section 4 we construct finite VTNCG's, some with the aid of this theorem and some without. ...
... This classic result of Whitney [Wh] for finite graphs has been generalized to infinite graphs in [I] and [Th1]. ... The set of faces of F will be denoted by F (F), and we will use the same symbols for both F and its planar ... Ends... more
... This classic result of Whitney [Wh] for finite graphs has been generalized to infinite graphs in [I] and [Th1]. ... The set of faces of F will be denoted by F (F), and we will use the same symbols for both F and its planar ... Ends and automorphisms of infinite graphs 397 embedding. ...
Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=dp(x,y) and aquasi-axis if lim... more
Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=dp(x,y) and aquasi-axis if lim infd(x,y)/dp(x,y)>0 asx, y range over the vertex set of p anddp(x,y)?8. The present paper brings together in greater generality results of R. Halin concerning
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
ABSTRACT Using the notion of fibers, where two rays belong to the same fiber if and only if they lie within bounded Hausdorff-distance of one another, we study how many fibers of a graph contain a geodetic ray and how many essentially... more
ABSTRACT Using the notion of fibers, where two rays belong to the same fiber if and only if they lie within bounded Hausdorff-distance of one another, we study how many fibers of a graph contain a geodetic ray and how many essentially distinct geodetic rays such “geodetic fibers” must contain. A complete answer is provided in the case of locally finite graphs that admit an almost transitive action by some infinite finitely generated, abelian group. Such graphs turn out to have either finitely many or uncountably many geodetic fibers. Furthermore, with finitely many possible exceptions, each of these fibers contains uncountably many geodetic rays. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 67–88, 2000
Research Interests:
Research Interests:
Research Interests:
A fiber in an infinite graph is an equivalence class of rays whereby two rays belong to the same fiber whenever each is contained in ann-neighborhood of the other for somen<∞. As this... more
A fiber in an infinite graph is an equivalence class of rays whereby two rays belong to the same fiber whenever each is contained in ann-neighborhood of the other for somen<∞. As this relation is a refinement of end-equivalence, it is of interest when applied to one-ended graphs, in particular to the class Ga, a*of one-ended, 3-connected, planar graphs whose valences
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
... theoreme de Jordan ainsi que le critere de Kuratowski), on demontre l'existence de suffisamment de chemins dans OZ pour que h ne puisse rester planaire, ce qui est absurde. Donc r=1, et l' est aretetransitif.... more
... theoreme de Jordan ainsi que le critere de Kuratowski), on demontre l'existence de suffisamment de chemins dans OZ pour que h ne puisse rester planaire, ce qui est absurde. Donc r=1, et l' est aretetransitif. La notion de classe terminale dans un graphe infini est due a Halin [4 ...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). If X is such a graph with vertex set V(X) and automorphism group A(X), we say that X is a graphical regular... more
In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). If X is such a graph with vertex set V(X) and automorphism group A(X), we say that X is a graphical regular representation (GRR) of a given abstract group G if (I) G ≅ A(X) , and (II) A(X) acts on V(X) as a regular permutation group; that is, given u, v ∈ V(X), there exists a unique φ ∈ A(X) for which φ(u) = v. That for any abstract group G there exists a graph X satisfying (I) is well-known (cf. [3]).
Research Interests:
1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-setand edge-set E(G(n, k)) to consist of all edges of the formwhere i is an integer. All subscripts in... more
1. Introduction. For integers n and k with 2 ≤ 2k < n, the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-setand edge-set E(G(n, k)) to consist of all edges of the formwhere i is an integer. All subscripts in this paper are to be read modulo n, where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2n, and G(5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)
Research Interests:
... References [1] L. :BABAI, Automorphism groups of graphs and edge-contraction. Discrete Math. 8, 13-=-20 (1974). ... Math. Ann. 202, 307--320 (1973). [10] HA Jv~c and ME WATKn~S, On the connectivites on finite and infinite graphs.... more
... References [1] L. :BABAI, Automorphism groups of graphs and edge-contraction. Discrete Math. 8, 13-=-20 (1974). ... Math. Ann. 202, 307--320 (1973). [10] HA Jv~c and ME WATKn~S, On the connectivites on finite and infinite graphs. Monatsh. 83, 121--131 (1977). Page 7. ...