Diameter-essential edges in a graph

The eccentricity e(v) of v is the distance to a farthest vertex from v. The diameter diam(G) is the maximum eccentricity among the vertices of G. The contraction of edge e=uv in G consists of removing e and identifying u and v as a single new vertex w, where w is adjacent to any vertex that at least one of u or v were adjacent to. The graph resulting from contracting edge e is denoted G/e. An edge e is diameter-essential if diam(G/e)<diam(G). Let c(G) denote the number of diameter-essential edges in graph G. In this paper, we study existence and extremal problems for c(G); determine bounds on c(G) in terms of diameter and order; and obtain characterizations of graphs achieving extreme values of c(G).     

Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V(G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is super-λ if and only if G\ncong K₂{G\ncong K_2}.     

Acyclic Edge Coloring of Planar Graphs Without Small Cycles

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. In this paper, it is shown that the conjecture holds for planar graphs without 4- and 5-cycles or without 4- and 6-cycles.     

An upper bound for the adjacent vertex distinguishing acyclic edge chromatic number of a graph

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′ _aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′ _aa (G) ≤ 32Δ. Supported by the Natural Science Foundation of Gansu Province (3ZS051-A25-025)     

Removable edges in cycles of a <Emphasis Type="Italic">k</Emphasis>-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N _G−e (x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4).     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

Bounded Direction–Length Frameworks

A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝ ^d for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G ⁺ be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G ⁺,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G ⁺,p). We use this to characterize when a mixed graph is generically bounded in ℝ ^d . As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded.     

Multigraphs with Δ ≥ 3 are Totally-(2Δ−1)-Choosable

The total graph T(G) of a multigraph G has as its vertices the set of edges and vertices of G and has an edge between two vertices if their corresponding elements are either adjacent or incident in G. We show that if G has maximum degree Δ(G), then T(G) is (2Δ(G) − 1)-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for Δ(G) > 3 was , by Borodin et al. When Δ(G) = 4, our algorithm gives a better upper bound. When Δ(G)∈{3,5,6}, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.).     

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).     

On the Structure of Contractible Edges in <Emphasis Type="Italic">k</Emphasis>-connected Partial <Emphasis Type="Italic">k</Emphasis>-trees

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k − 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees.     

The k-Restricted Edge Connectivity of Balanced Bipartite Graphs

For a connected graph G = (V, E), an edge set S ì E{S\subset E} is called a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λ _k (G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex sets U₁,U₂ ì V(G){U_1,U_2\subset V(G)}, denote the set of edges of G with one end in U ₁ and the other in U ₂ by [U ₁, U ₂]. Define x_k(G)=min{|[U,V(G)\ U]|: U{\xi_k(G)=\min\{|[U,V(G){\setminus} U]|: U} is a vertex subset of order k of G and the subgraph induced by U is connected}. A graph G is said to be λ _k -optimal if λ _k (G) = ξ _k (G). A graph is said to be super-λ _k if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. In this paper, we present some degree-sum conditions for balanced bipartite graphs to be λ _k -optimal or super-λ _k . Moreover, we demonstrate that our results are best possible.     

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood N _G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values , taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γ _st ′(G). Obviously, γ _st ′(G) is defined only for graphs G which have no connected components isomorphic to K ₂. In this paper we present some lower bounds for γ _st ′(G). In particular, we prove that γ _st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ _st ′(T). Research supported by a Faculty Research Grant, University of West Georgia.     

New results on 3-domination critical graphs

A graph G is said to be 3-domination critical if its domination number γ(G) = 3 and γ(G + e) = 2 for any edge e not contained in G. In this paper we first establish some structural properties of 3-domination critical graphs with diameter equal to 3. In particular, this allows us to characterize a special family of 3-domination critical graphs which contains those with minimum degree one. Moreover, we show that if the minimum degree of a 3-domination critical graph G is at least 3, then α(G) ≤ κ(G) + 1 or G is superconnected, where α(G) is the independence number and κ(G) is the vertex-connectivity of G.     

Bounds on the Total Restrained Domination Number of a Graph

Let G = (V, E) be a graph. A set S í V{S \subseteq V} is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ _tr (G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ _tr (G) ≤ n − diam(G) − r + 1.     

On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by x′ _Aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.     

Some Notes on Signed Edge Domination in Graphs

The closed neighborhood N_G[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ_s^′(G) of G is defined as . Recently, Xu proved that γ_s^′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ_s^′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ_s^′(G) = 1 − k, 2 − k. A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia. Send offprint requests to: Abdollah Khodkar.     

Paired-Domination Subdivision Numbers of Graphs

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ _pr (G), is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number sd _γpr (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. In this paper we establish upper bounds on the paired-domination subdivision number and pose some problems and conjectures.     

On Fixed Edges and Edge-Reconstruction of Series-Parallel Networks

 An edge e of a graph G is said to be a fixed edge if G−e+e ^′≅G implies that e ^′=e, and a forced edge if G−e+e ^′ is an edge-reconstruction of G implies that e ^′=e. In this paper, we use the method of excludable configurations to investigate the fixed edges and the forced edges of series-parallel networks. It is proved that all series-parallel networks contain fixed edges except P ₃∨K ₁ and P ₄∨K ₁, and that all series-parallel networks are edge-reconstructible. Received: December 22, 1997 Final version received: July 21, 1999     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.