We introduce the concept of a constrained pointed pseudo-triangulation TG of a point set S with respect to a pointed planar straight line graph G = (S,E). For the case that G forms a simple polygon P with vertex set S we give tight bounds on the vertex degree of TG.

A construction based on the Biggs-Smith graph is shown to produce an edge-disjoint union of 102 copies of K4 forming a {K4, L(Q3)}K3 -ultrahomogeneous graph dressed as the non-line-graphical Menger graph of a self-dual (1024)-configuration. This stands in contrast to the selfdual (424)-configuration of [8], whose Menger graph is a non-line-graphical {K4, K2,2,2}K2 -ultrahomogeneous graph.

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In [J. Graph Theory, 11 (1987), 399-407], Broersma and Veldman proposed an open problem: For a given positive integer k, determine the value s for which the statement Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected is...

An on-line vertex coloring algorithm receives the vertices of a graph in some externally determined order, and, whenever a new vertex is presented, the algorithm also learns to which of the previously presented vertices the new vertex is adjacent. As each vertex is received, the algorithm must make an irrevocable choice of a color to assign the new vertex, and it makes this choice without knowl...

A graph G is a quasi-line graph if for every vertex v ∈ V (G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi-line graphs is a proper superset of the class of line graphs. Hadwiger’s conjecture states that if a graph G is not t-colorable then it contains Kt+1 as a minor. This conjecture has been proved for line graphs by Reed and Seymour [10]. We...

A graph G is a quasi-line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques. The class of quasi-line graphs is a proper superset of the class of line graphs. A theorem of Shannon’s implies that if G is a line graph then it can be properly colored using no more than 32ω(G) colors, where ω(G) is the size of the largest clique in G. In this paper w...

We prove that for every d ≥ 3 the homomorphism order of the class of line graphs of finite graphs with maximal degree d is universal. This means that every finite or countably infinite partially ordered set may be represented by line graphs of graphs with maximal degree d ordered by the existence of a homomorphism.

Let X be a simple connected graph with n vertices and m edges. Every vertex of X is assigned either black state or white state. We move by selecting a vertex v of X having black state and then change the states of all neighbors of v. This is the flipping puzzle on X and it corresponds to a group action. We referred this group to the flipping group of X. In this paper, we are mainly concerned ab...