Diameter-vital edges in a graph

The graph resulting from contracting edge e is denoted as G/e and the graph resulting from deleting edge e is denoted as G − e. An edge e is diameter-essential if diam(G/e) < diam(G), diameter-increasing if diam(G − e) < diam(G), and diameter-vital if it is both diameter-essential and diameter-increasing. We partition the edges that are not diameter-vital into three categories. In this paper, we study realizability questions relating to the number of edges that are not diameter-vital in the three defined categories. A graph is diameter-vital if all its edges are diameter-vital. We give a structural characterization of diameter-vital graphs.     

Coloring the edges of a directed graph

Finding large cliques in a graph is an important problem in applied discrete mathematics. In directed graph a possible corresponding problem is finding large transitive subtournaments. It is well-known that coloring the graph speeds up the clique search in the undirected case. In this paper we propose coloring schemes to speed up the tournament search in the directed case. We prove two complexity results about the proposed colorings. A consequence of these results is that in practical computations we have to be content with approximate greedy coloring algorithms.     

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).     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Partitioning a graph into two square-cycles

A square-cycle is the graph obtained from a cycle by joining every pair of vertices of distance two in the cycle. The length of a square-cycle is the number of vertices in it. Let G be a graph on n vertices with minimum degree at least 2/3n and let c be the maximum length of a square-cycle in G. Pósa and Seymour conjectured that c = n. In this paper, it is proved that either c = n or 1/2n ≤ c ≤ 2/3n. As an application of this result, it is shown that G has two vertex-disjoint square-cycles C₁, and C₂ such that V(G) = V(C₁) ∪ V(C₂). © 1996 John Wiley & Sons, Inc.     

Clique coverings of the edges of a random graph

The edges of the random graph (with the edge probabilityp=1/2) can be covered usingO(n ²lnlnn/(lnn)²) cliques. Hence this is an upper bound on the intersection number (also called clique cover number) of the random graph. A lower bound, obtained by counting arguments, is (1–)n ²/(2lgn)².Research supported in part by ONR Grant N00014-85K0570 and by NSA/MSP Grant MDA904-90-H-4011.     

Further remarks on partitioning the edges of a graph

The main result is that if m and k are odd integers with m ≥ k ≥ 1, then any graph which is the union of m graphs of maximum valence k is also the union of k graphs of maximum valence m. This is not generally true if k > m.     

Non-contractible edges in a 3-connected graph

An edgee in a 3-connected graphG is contractible if the contraction ofe inG results in a 3-connected graph; otherwisee is non-contractible. In this paper, we prove that the number of non-contractible edges in a 3-connected graph of orderp≥5 is at most $$3p - \left[ {\frac{3}{2}(\sqrt {24p + 25} - 5} \right],$$ and show that this upper bound is the best possible for infinitely many values ofp.     

Crossing-critical edges and Kuratowski subgraphs of a graph

An edge e of a graph G is said to be crossing-critical if cr(G ? e) < cr(G), where cr(G) denotes the crossing number of G on the plane. It is proved that any crossing-critical edge e of a graph G for which cr(G ? e) ≤ 1 belongs to a subdivision of K₅ or K_{3, 3}, the Kuratowski subgraphs of G. Further, as regards crossing-critical edges e of G for which cr(G ? e) ≥ 5, it is shown that the properties of “being a crossing-critical edge of G” and “being contained in a Kuratowski subgraph of G” are independent.     

Partitioning a graph into alliance free sets

A strong defensive alliance in a graph G=(V,E) is a set of vertices A⊆V, for which every vertex v∈A has at least as many neighbors in A as in V−A. We call a partition A,B of vertices to be an alliance-free partition, if neither A nor B contains a strong defensive alliance as a subset. We prove that a connected graph G has an alliance-free partition exactly when G has a block that is other than an odd clique or an odd cycle.     

Partitioning a graph into monochromatic connected subgraphs

We show that every -edge-colored graph on vertices with minimum degree at least can be partitioned into two monochromatic connected subgraphs, provided is sufficiently large. This minimum degree condition is tight and the result proves a conjecture of Bal and DeBiasio. We also make progress on another conjecture of Bal and DeBiasio on covering graphs with large minimum degree with monochromatic components of distinct colors.     

Covering the edges of a graph by three odd subgraphs

We prove that any finite simple graph can be covered by three of its odd subgraphs, and we construct an infinite sequence of graphs where an edge‐disjoint covering by three odd subgraphs is not possible. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 77–82, 2006     

Removable edges in a spanning tree of a k-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G O e denotes the graph obtained from G by the following way： deleting e to get G - e, and for any end vertex of e with degree k - 1 in G - e, say x, deleting x, and then adding edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of k-connected graphs have been investigated. In the present paper, we investigate the distribution of removable edges on a spanning tree of a k-connected graph （k ≥ 4）.     

The minimum number of triangles covering the edges of a graph

Suppose that a connected graph G has n vertices and m edges, and each edge is contained in some triangle of G. Bounds are established here on the minimum number t_min(G) of triangles that cover the edges of G. We prove that ?(n - 1)/2? ? t_min(G) with equality attained only by 3-cactii and by strongly related graphs. We obtain sharp upper bounds: if G is not a 4-clique, then. The triangle cover number t_min(G) is also bounded above by Γ(G) = m - n + c, the cyclomatic number of a graph G of order n with m edges and c connected components. Here we give a combinatorial proof for t_min(G) ? Γ(G) and characterize the family of all extremal graphs satisfying equality.     

On the number of edges in the transitive closure of a graph

A property of the square sum of partitions of integers is investigated. The square sum has a direct relation to the number of edges in the transitive closure of a graph. This paper is concerned with the problem of determining the minimum missing value in the sequence of square sums. Asymptotically tight lower and upper bounds on this value are obtained. A consequence of the main result for closure size prediction is also mentioned.     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.