Dense graphs and edge reconstruction

By a well-known result of Nash-Williams if a graphG is not edge reconstructible, then for all ,|A||E(G)| mod 2 we have a permutation ofV(G) such thatE(G)E(G)=A. Here we construct infinitely many graphsG having this curious property and more than edges.Research (partially) supported by Hungarian National Foundation for Scientific Research Grant No.T016389.     

The edge reconstruction of hamiltonian graphs

If a graph G on n vertices contains a Hamiltonian path, then G is reconstructible from its edge-deleted subgraphs for n sufficiently large.     

List-edge and list-total colorings of graphs embedded on hyperbolic surfaces

In the paper, we prove that if G is a graph embeddable on a surface of Euler characteristic ε<0 and , then and . This extends a result of Borodin, Kostochka and Woodall [O.V. Borodin, A.V. Kostochka, D.R. Woodall, List-edge and list-total colorings of multigraphs, J. Comb. Theory Series B 71 (1997) 184-204].     

On the Kneser embedded edge theorem

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 145–147, December, 1991.     

Counterexamples to the edge reconstruction conjecture for infinite graphs

For every infinite cardinal α, there exists a graph with α edges which is not uniquely reconstructible from its family of edge-deleted subgraphs.     

On the edge multicoloring of unicyclic graphs

A multicoloring of an edge weighted graph is an assignment of intervals to its edges so that the intervals of adjacent edges do not intersect at interior points and the length of each interval is equal to the weight of the edge. The minimum length of the union of all intervals is called an edge multichromatic number of the graph. The maximum weighted degree of a vertex (i.e., the sum of the weights of all edges incident with it) is an evident lower bound of this number. There are available the examples in which the multichromatic number is one and a half times larger than the lower bound. Also, there is a conjecture that the bound cannot exceeded by a larger factor. Here we prove this conjecture for the class of unicyclic graphs.     

3‐Coloring graphs embedded in surfaces

In this article, we show that there exists an integer k(Σ) ≥ 5 such that, if G is a graph embedded in a surface Σ without i‐circuits, 4 ≤ i ≤ k(Σ), then G is 3‐colorable. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 140–143, 2000     

On edge domination numbers of graphs

Let and be the signed edge domination number and signed star domination number of G, respectively. We prove that holds for all graphs G without isolated vertices, where n=|V(G)|?4 and m=|E(G)|, and pose some problems and conjectures.     

On acyclic colorings of graphs on surfaces

A properk-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring. This paper shows that the acyclic chromatic number of the projective plane is at most 7. The acyclic chromatic number of an arbitrary surface with Euler characteristic η=−γ is at mostO(γ^4/7). This is nearly tight; for every γ>0 there are graphs embeddable on surfaces of Euler characteristic −γ whose acyclic chromatic number is at least Ω(γ^4/7/(logγ)^1/7). Therefore, the conjecture of Borodin that the acyclic chromatic number of any surface but the plane is the same as its chromatic number is false for all surfaces with large γ (and may very well be false for all surfaces). This author's research was supported in part by a United States Israeli BSF grant. This author's research was supported by the Ministry of Research and Technology of Slovenia, Research Project P1-0210-101-92. This author's research was supported by the Office of Naval Research, grant number N00014-92-J-1965.     

On the edge Szeged index of bridge graphs

In this Note, we introduce a formula for the edge Szeged index of bridge graphs. Using this formula, the edge Szeged indices of several graphs are computed.     

On the signed edge domination number of graphs

Let be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n≥2), . In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that . Also for every two natural numbers m and n, we determine , where K_m,n is the complete bipartite graph with part sizes m and n.     

On dihedral flows in embedded graphs

Let be a multigraph with for each vertex a cyclic order of the edges incident with it. For , let be the dihedral group of order . Define . Goodall et al in 2016 asked whether admits a nowhere-identity -flow if and only if it admits a nowhere-identity -flow with (a “nowhere-identity dihedral -flow”). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere-identity -flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true are described. We focus particularly on cubic graphs.     

Total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

Let G be a graph which can be embedded in a surface of nonnegative Euler characteristic.In this paper,it is proved that the total chromatic number of G is △(G)+1 if △(G)9,where △(G)is the maximum degree of G.     

On the weak reconstruction of Cartesian-product graphs

In this paper we reconstruct nontrivial connected Cartesian-product graphs from single vertex deleted subgraphs. We show that all one-vertex extensions of a given connected graph H, finite or infinite, to a nontrivial Cartesian product are isomorphic.     

On optimizing edge connectivity of product graphs

This work studies the super edge connectivity and super restricted edge connectivity of direct product graphs, Cartesian product graphs, strong product graphs and lexicographic product graphs. As a result, sufficient conditions for optimizing the edge connectivity and restricted edge connectivity of these graphs are presented.     

On edge colorings of 1-toroidal graphs

A graph is 1-toroidal, if it can be embedded in the torus so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-toroidal graph with maximum degree Δ≥ 10 is of class one in terms of edge coloring. Meanwhile, we show that there exist class two 1-toroidal graphs with maximum degree Δ for each Δ≤ 8.     

On the 3-restricted edge connectivity of permutation graphs

An edge cut W of a connected graph G is a k-restricted edge cut if G−W is disconnected, and every component of G−W has at least k vertices. The k-restricted edge connectivity is defined as the minimum cardinality over all k-restricted edge cuts. A permutation graph is obtained by taking two disjoint copies of a graph and adding a perfect matching between the two copies. The k-restricted edge connectivity of a permutation graph is upper bounded by the so-called minimum k-edge degree. In this paper some sufficient conditions guaranteeing optimal k-restricted edge connectivity and super k-restricted edge connectivity for permutation graphs are presented for k=2,3.