Let G be a graph. A G-trade of volume m is a pair (T ,T ′), where each of T and T ′ consists of m graphs, pairwise edge-disjoint, isomorphic to G, such that T ∩T ′ = ∅ and the union of the edge sets of the graphs in T is identical to the union of the edge sets of the graphs in T ′. Let X(G) be the set of non-negative integers m such that no G-trade of volume m exists. In this paper we prove tha...

Let G be a graph. A G-trade of volume m is a pair (T , T ′), where each of T and T ′ consists of m graphs, pairwise edge-disjoint, isomorphic to G, such that T ∩T ′ = ∅ and the union of the edge sets of the graphs in T is identical to the union of the edge sets of the graphs in T ′. Let X(G) be the set of non-negative integers m such that no G-trade of volume m exists. In this paper we prove th...

a graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ e(g)} consists of n integers. inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of g and k2 , and the disjoint :union: of g...

A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. Here we give a construction for all quasi-line graphs; it turns out that there are basically two kinds of connected quasi-line graphs, one a generalization of line graphs, and the other ...

for $q in {7,8,9,11,13,16}$, we consider the primitive actions of $l_2(q)$ and use key-moori method 1 as described in [codes, designs and graphs from the janko groups {$j_1$} and{$j_2$}, {em j. combin. math. combin. comput.}, {bf 40} (2002) 143--159., correction to: ``codes, designs and graphs from the janko groups{$j_1$} and {$j_2$}'' [j. combin. math. combin. comput. {bf 40} (2002) 143--159],...

A connected edge-disjoint union Y of 102 tetrahedra (copies ofK4) is constructed which is Menger graph of a self-dual (1024)-configuration and K4-ultrahomogeneous. As Y is not a line graph, we ask whether there exists a non-line-graphical K4-ultrahomogeneous Menger graph of a self-dual (n4)-configuration which is connected edge-disjoint union of n copies of K4, for n / ∈ {42, 102}. Moreover, Y ...

Let G be a graph with domination number γ(G). A dominating set S ⊆ V (G) has property UK if all components of the subgraph it induces in G are complete. The union of complete graphs domination number of a graph G, denoted γuk(G), is the minimum possible size of a dominating set of G, which has property UK. Results on changing and unchanging of γuk after vertex removal are presented. Also forbid...

The graph compositions‘ notion was introduced by A. Knopfmacher and M. E. Mays [1]. In this note we add to these a new construction of tree-like graphs where nodes are graphs themselves. The first examples of these tree-like compositions, a corresponding theorem and resulting conclusions are provided. AMS Classification Numbers: 05A18, 05C05, 05C30, 11B37.