Garry L. Johns

Boundary vertices in graphs

The distance d(u, v) between two vertices u and v in a nontrivial connected graph G is the length of a shortest u-v path in G. For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. A vertex v of G is a peripheral vertex if e(v) is the diameter of G. The subgraph of G induced by its peripheral vertices is the periphery Per(G) of G. A vertex u of G is an eccentric vertex of a vertex v if d(u, v)= e(v). A vertex x is an eccentric vertex of G if x is an eccentric vertex of some vertex of G. The subgraph of G induced by its eccentric vertices is the eccentric subgraph Ecc(G) of G. A vertex u of G is a boundary vertex of a vertex v if d(w,v) ≤ d(u,v) for all w ∈ N(u). A vertex u is a boundary vertex of G if u is a boundary vertex of some vertex of G. The subgraph of G induced by its boundary vertices is the boundary ∂(G) of G. A graph H is a boundary graph if H = ∂(G) for some graph G. We study the relationship among the periphery, eccentric subgraph, and boundary of a connected graph and establish a characterization of all boundary graphs. It is shown that per each triple a, b, c of integers with 2 ≤ a ≤ b ≤ c, there is a connected graph G such that Per(G) has order a, Ecc(G) has order b, and ∂(G) has order c. Moreover, for each triple r,s,t of rational numbers with 0 < r ≤ s ≤ t ≤ 1, there is a connected graph G of order n such that |V(Per(G))|/n=r, |V(Ecc(G))|/n = s, and |V(∂(G))| n=t.

Directed distance in digraphs: centers and medians

The directed distance dD(u, v) from a vertex u to a vertex v in a strong digraph D is the length of a shortest (directed) u - v path in D. The eccentricity of a vertex v in D is the directed distance from v to a vertex furthest from v. The distance of a vertex v in D is the sum of the directed distances from v to the vertices of D. The center C(D) of D is the subdigraph induced by those vertices of minimum eccentricity, while the median M(D) of D is the subdigraph induced by those vertices of minimum distance. It is shown that for every two asymmetric digraphs D1 and D2, there exists a strong asymmetric digraph H such that C(H) ≅ D1 and M(H) ≅ D2, and where the directed distance from C(H) to M(H) and from M(H) to C(H) can be arbitrarily prescribed. Furthermore, if K is a nonempty asymmetric digraph isomorphic to an induced subdigraph of both D1 and D2, then there exists a strong asymmetric digraph F such that C(F) ≅ D1, M(F) ≅ D2 and C(F) ∩ M(F) ≅ K.

Detour Distance in Graphs

Abstract For vertices u and v in a connected graph G, the detour distance d* (u, v) between u and v is the length of a longest path P for which the subgraph induced by the vertices of P is P itself. A graph G is called a detour graph if d* (u, v) equals the standard distance between u and v in G for every pair u, v of vertices of G. Several results concerning detour distance and detour graphs are presented.

Knight’s Tours on Rectangular Chessboards Using External Squares

The classic puzzle of finding a closed knight’s tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight’s tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight’s tour. In this paper we define an extended closed knight’s tour for a rectangular chessboard as a closed knight’s tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight’s tour?

HAMILTONIAN-COLORED POWERS OF STRONG DIGRAPHS

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power D k of D is that digraph having vertex set V (D) with the property that (u,v) is an arc of D k if the directed distance ~ dD(u,v) from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph D k is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph D k is distance-colored if each arc (u,v) of D k is assigned the color i where i = ~ dD(u,v). The digraph D k is Hamiltonian-colored if D k contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which D k is Hamiltonian-colored is the Hamiltonian coloring exponent hce(D) of D. For each integer n ≥ 3, the Hamiltonian coloring exponent of the directed cycle ~ Cn of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph Dk such that hce(Dk) = k with the added property that every properly colored Hamiltonian cycle in the kth power of Dk must use all k colors. It is shown for every positive integer p there exists a a connected graph G with two different strong orientations D

A Sharp Bound for the Marginal Appendage Number

Abstract For a connected graph G , the distance d(u, v) between two vertices u and v is the length of a shortest u − v path in G and the distance d(v) of a vertex v is the sum of the distances between v and all vertices of G . The margin, μ (G) , is the subgraph induced by vertices of G having the maximum distance. It is known that every graph is isomorphic to the margin of some graph H . For a graph G , the marginal appendage number is defined as min { p ( H ) − p ( G ) ∣ μ ( H ) = G }. In this paper it is shown that Δ ( G ) + 2 is a sharp bound for the marginal appendage number.

S-Distance in Trees

For a connected graph G, a subset S of V(G) and vertices u, v of G, the S-distance ds(u,v) from u to v is the length of a shortest u — v walk in G that contains every vertex of S. The S-eccentricity es(v) of a vertex v is the maximum S-distance from v to each vertex of G and the S-diameter diamsG is the maximum S-eccentricity among the vertices of G. For a tree T, a formula is given for ds(u,v) and for S ≠ ϕ, it is shown that diamsT is even. Finally, the complexity of finding ds(u,v) is discussed.