Claw-free graphs are edge reconstructible

The Edge Reconstruction Conjecture states that all graphs with at least four edges are determined by their edge-deleted subgraphs. We prove that this is true for claw-free graphs, those graphs with no induced subgraph isomorphic to K_1,3. This includes line graphs as a special case.     

Nearly acyclic graphs are reconstructible

We prove that a graph G is reconstructible if G has a node v with G-v acyclic. The proof uses colored graphs and shows how to reconstruct some graphs from pieces which share a common subgraph having few automorphisms.     

The reconstruction conjecture is true if all 2-connected graphs are reconstructible

It is shown that the Reconstruction Conjecture is true for all finite graphs if it is true for the 2-connected ones.     

One edge union of shell graphs and one vertex union of complete bipartite graphs are cordial

In this paper we prove that one edge union of k-copies of shell graphs H(n,n−3) is cordial, for all n⩾4 and k⩾1 and one vertex union of t copies of complete bipartite graph K_m,n is cordial.     

New edge neighborhood graphs

Let G be an undirected simple connected graph, and e = uv be an edge of G. Let N _G(e) be the subgraph of G induced by the set of all vertices of G which are not incident to e but are adjacent to u or v. Let N _e be the class of all graphs H such that, for some graph G,N _G (e) H for every edge e of G. Zelinka [3] studied edge neighborhood graphs and obtained some special graphs in N _e. Balasubramanian and Alsardary [1] obtained some other graphs in N _e. In this paper we given some new graphs in N _e.     

Semi-hyper-connected edge transitive graphs

A graph G is said to be hyper-connected if the removal of every minimum cut creates exactly two connected components, one of which is an isolated vertex. In this paper, we first generalize the concept of hyper-connected graphs to that of semi-hyper-connected graphs: a graph G is called semi-hyper-connected if the removal of every minimum cut of G creates exactly two components. Then we characterize semi-hyper-connected edge transitive graphs.     

Cubic graphs with three Hamiltonian cycles are not always uniquely edge colorable

The generalized Petersen graph P(6k + 3, 2) has exactly 3 Hamiltonian cycles for k ≥ 0, but for k ≥ 2 is not uniquely edge colorable. This disproves a conjecture of Greenwell and Kronk [1].     

Super-connected edge transitive graphs

A graph G is said to be super-connected if any minimum cut of G isolates a vertex. In a previous work due to the second author of this note, super-connected graphs which are both vertex transitive and edge transitive are characterized. In this note, we generalize the characterization to edge transitive graphs which are not necessarily vertex transitive, showing that the only irreducible edge transitive graphs which are not super-connected are the cycles C_n(n?6) and the line graph of the 3-cube, where irreducible means the graph has no vertices with the same neighbor set. Furthermore, we give some sufficient conditions for reducible edge transitive graphs to be super-connected.     

Semistrong edge coloring of graphs

Weakening the notion of a strong (induced) matching of graphs, in this paper, we introduce the notion of a semistrong matching. A matching M of a graph G is called semistrong if each edge of M has a vertex, which is of degree one in the induced subgraph G[M]. We strengthen earlier results by showing that for the subset graphs and for the Kneser graphs the sizes of the maxima of the strong and semistrong matchings are equal and so are the strong and semistrong chromatic indices. Similar properties are conjectured for the n‐dimensional cube. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 39–47, 2005     

Acyclic edge colorings of graphs

A proper coloring of the edges of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

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.     

Orientable edge colorings of graphs

An edge coloring of a graph is orientable if and only if it is possible to orient the edges of the graph so that the color of each edge is determined by the head of its corresponding oriented arc. The goals of this paper include finding a forbidden substructure characterization of orientable colorings and giving a linear time recognition algorithm for orientable colorings.An edge coloring is lexical if and only if it is possible to number the vertices of the graph so that the color of each edge is determined by its lower endpoint. Lexical colorings are, of course, the orientable colorings in which the underlying orientation is acyclic. Lexical colorings play an important role in Canonical Ramsey theory, and it is this standpoint that motivates the current study.     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

Regular graphs with high edge degree

For any vertex x of a graph G let Δ(x) denote the set of vertices adjacent to x. We seek to describe the connected graphs G which are regular of valence n and in which for all adjacent vertices x and y |Δ(x) ∩ Δ(y)| = n ? 1 ? s. It is known that the complete graphs are the graphs for which s = 0. For any s, any complete many-partite graph, each part containing s + 1 vertices, is such a graph. We show that these are the only such graphs for which the valence exceeds 2s² ? s + 1. The graphs satisfying these conditions for s = 1 or 2 are characterized (up to the class of trivalent triangle-free graphs.)     

Minimally 3-restricted edge connected graphs

For a connected graph G=(V,E), an edge set S⊂E is a 3-restricted edge cut if G−S is disconnected and every component of G−S has order at least three. The cardinality of a minimum 3-restricted edge cut of G is the 3-restricted edge connectivity of G, denoted by λ₃(G). A graph G is called minimally 3-restricted edge connected if λ₃(G−e)<λ₃(G) for each edge e∈E. A graph G is λ₃-optimal if λ₃(G)=ξ₃(G), where , ω(U) is the number of edges between U and V?U, and G[U] is the subgraph of G induced by vertex set U. We show in this paper that a minimally 3-restricted edge connected graph is always λ₃-optimal except the 3-cube.     

Adjacent strong edge coloring of graphs

For a graph G(V, E), if a proper k-edge coloring ƒ is satisfied with C(u) ≠ C(v) for uv ∈ E(G), where $\text{C(u) = {ƒ(uv) | uv ∈ E}}$, then ƒ is called k-adjacent strong edge coloring of G, is abbreviated k-ASEC, and χ′_as(G) = min{k | k-ASEC of G} is called the adjacent strong edge chromatic number of G. In this paper, we discuss some properties of χ′_as(G), and obtain the χ′_as(G) of some special graphs and present a conjecture: if G are graphs whose order of each component is at least six, then χ′_as(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree of G.     

图的符号边控制数

图的符号边控制数有着许多重要的应用背景.已知它的计算是NP-完全问题,因而确定其精确值有重要意义.本文确定了图F*n+1、H n和P*n的符号边控制数.