Leonid S. Melnikov

Some counterexamples associated with the three-color problem

Abstract In this paper we construct some counterexamples of non-3-colorable planar graphs, using the notion of “quasi-edges”. The minimality of some quasi-edges is proved.

Some results on the Wiener index of iterated line graphs

Abstract The Wiener index W ( G ) of a graph G is the sum of distances between all unordered pairs of vertices. This notion was motivated by various mathematical properties and chemical applications. For a tree T, it is known that W ( T ) and W ( L ( T ) ) are always distinct. It is shown that there is an infinite family of trees with W ( T ) = W ( L 2 ( T ) ) .

The edge chromatic number of a directed-mixed multigraph

It was conjectured in [Wang, to appear in The Australasian Journal of Combinatorics] that, for each integer k ≥ 2, there exists M(k) such that if G= (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ M(k) and d(x) + d(y) ≥ n + k for each pair of nonadjacent vertices x and y of G with x e V1 and y e V2, then, for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei e E(Ci for all i e {1, …, k} and V(C1 ⊎ ··· ∪ Ck) = V(G). This conjecture is also verified for k = 2, 3 in [Wang, to appear; Wang, manuscript]. In this article, we prove this conjecture to be true if n ≥ 3k, i.e., M(k) ≤ 3k. We will also show that, if n ≥ 3k, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck of length at most 6 in G such that ei e E(Ci) for all i e {1, …, k}.

Deeply asymmetric planar graphs

It is proved that by deleting at most 5 edges every planar (simple) graph of order at least 2 can be reduced to a graph having a non-trivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4.

Regular integral sum graphs

Abstract Given a set of integers S, G(S)=(S,E) is a graph, where the edge uv exists if and only if u + v ∈ S . A graph G =( V , E ) is an integral sum graph or ISG if there exists a set S ⊂ Z such that G = G ( S ). This set is called a labeling of G . The main results of this paper concern regular ISGs. It is proved that all 2-regular graphs with the exception of C 4 are integral sum graphs and that for every positive integer r there exists an r -regular ISG.

Counterexamples to Grötzsch-Sachs-Koester's conjecture

Let G be a 4-regular planar graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly on cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Grotzsch-Sachs-Koesters conjecture states that if the cycles of G can be partitioned into four classes, such that two cycles in the same classes are disjoint, G is vertex 3-colorable. In this note, the conjecture is disproved.

Wiener index of generalized stars and their quadratic line graphs

The Wiener index, W , is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of ∆ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W (S) = W (L(L(S)) exist only for 4 ≤ ∆ ≤ 6. Infinite families of generalized stars with this property are constructed.

Note: 4-chromatic edge critical Grötzsch-Sachs graphs

Let G be a 4-regular plane graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly one cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Two 4-chromatic edge critical graphs of order 48 generated by four curves are presented.

Vertex-oblique graphs

Let x be a vertex of a simple graph G. The vertex-type of x is the lexicographically ordered degree sequence of its neighbors. We call the graph G vertex-oblique if there are no two vertices in V(G) which are of the same vertex-type. We will show that the set of vertex-oblique graphs of arbitrary connectivity is infinite.

On 4-chromatic edge-critical regular graphs of high connectivity

New examples of 4-chromatic edge-critical r-regular and r-connected graphs are presented for r = 6,8,10.