Donald W. VanderJagt

Recognizable colorings of graphs

Let G be a connected graph and let c : V (G) ! f1; 2; : : : ; kg be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code(v) = (a0; a1; : : : ; ak) where a0 is the color assigned to v and for 1 i k, ai is the number of vertices adjacent to v that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number rn(G) of G is the minimum positive integer k for which G has

Graphs with prescribed local connectivities

A graph G is locally n-connected (locally n-edge connected) if the neighborhood of each vertex of G is n-connected (n-edge connected). The local connectivity (local edge-connectivity) of G is the maximum n for which G is locally n-connected (locally n-edge connected). It is shown that if k and m are integers with O =< k < m, then a graph exists which has connectivity m and local connectivity k. Furthermore, such a graph with smallest order is determined. Corresponding results are obtained involving the local connectivity and the local edge-conectivity.

On γ-labelings of trees

Let G be a graph of order n and size m. A γ-labeling of G is a oneto-one function f : V (G) → {0, 1, 2, . . . , m} that induces a labeling f ′ : E(G) → {1, 2, . . . ,m} of the edges of G defined by f ′(e) = |f(u)−f(v)| for each edge e = uv of G. The value of a γ-labeling f is val(f) = ∑ e∈E(G) f ′(e). The maximum value of a γ-labeling of G is defined as valmax(G) = max{val(f) : f is a γ-labeling of G}; while the minimum value of a γ-labeling of G is valmin(G) = min{val(f) : f is a γ-labeling of G}. 364 G. Chartrand, D. Erwin, D.W. VanderJagt and P. Zhang The values valmax(Sp,q) and valmin(Sp,q) are determined for double stars Sp,q. We present characterizations of connected graphs G of order n for which valmin(G) = n or valmin(G) = n + 1.

On the planarity of jump graphs

Abstract For a graph G of size m⩾1 and edge-induced subgraphs F and H of size k ( 1⩽k⩽m ), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uv∈E(F) , wx∈E(G)−E(F) , and H=F−uv+wx . The minimum number of edge jumps required to transform F into H is the k -jump distance from F to H . For a graph G of size m⩾1 and an integer k with 1⩽k⩽m , the k -jump graph J k (G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J k (G) are adjacent if and only if the k -jump distance between the corresponding subgraphs is 1. All connected graphs G for which J 2 (G) is planar are determined.

On the planarity of iterated jump graphs

Abstract For a graph G of size m⩾1 and edge-induced subgraphs F and H of size r ( 1⩽r⩽m ), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uv∈E(F) , wx∈E(G)−E(F) , and H=F−uv+wx . The minimum number of edge jumps required to transform F into H is the jump distance from F to H . For a graph G of size m⩾1 and an integer r with 1⩽r⩽m , the r -jump graph J r (G) is that graph whose vertices correspond to the edge-induced subgraphs of size r of G and where two vertices of J r (G) are adjacent if and only if the jump distance between the corresponding subgraphs is 1. For k⩾2 and r⩾1 , the k th iterated r -jump graph J r k (G) is defined as J r (J r k−1 (G)) , where J r 1 (G)=J r (G) . An infinite sequence {G i } of graphs is planar if every graph G i is planar. All graphs G for which {J r k (G)} ( r=1,2 ) is planar are determined and it is shown that if the sequence {J r k (G)} is nonplanar, then lim k→∞ gen( J r k (G))=∞ , where gen (G) denotes the genus of a graph G .