Partitioning the edges of a graph

For any integer m (≥2), it is known that there are simple graphs of maximum valence m whose edges cannot be coloured with m colours in such a way that adjacent edges shall have different colours. We find those values of m and k for which it is true that every simple graph whose maximum valence does not exceed mk can be coloured with m colours in such a way that no colour appears more than k times at any vertex.     

Coloring the square of a planar graph

We prove that for any planar graph G with maximum degree Δ, it holds that the chromatic number of the square of G satisfies χ(G²) ≤ 2Δ + 25. We generalize this result to integer labelings of planar graphs involving constraints on distances one and two in the graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 110–124, 2003     

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.     

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).     

Coloring edges of embedded graphs

In this paper, we prove that any graph G with maximum degree , which is embeddable in a surface Σ of characteristic χ(Σ) ≤ 1 and satisfies , is class one. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 197–205, 2000     

Coloring graph bundles

Graph bundles generalize the notion of covering graphs and products of graphs. Several results about the chromatic numbers of graph bundles based on the Cartesian product, the strong product and the tensor product are presented. © 1995 John Wiley & Sons, Inc.     

Disjoint directed quadrilaterals in a directed graph

Let D be a directed graph of order 4k, where k is a positive integer. Suppose that the minimum degree of D is at least 6k ? 2. We show that D contains k disjoint directed quadrilaterals with only one exception. © 2005 Wiley Periodicals, Inc. J Graph Theory     

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.     

Coloring uniform hypergraphs with few edges

A hypergraph is b‐simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r‐uniform non‐t‐colorable hypergraph of girth at least g. Erd?s and Lovász proved that A result of Szabó improves the lower bound by a factor of r^2?? for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b‐simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and ? > 0 and sufficiently large r, every r‐uniform b‐simple hypergraph with maximum edge‐degree at most t^rr^1?? is t‐colorable. Some results hold for list coloring, as well. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

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.     

Packing rooted directed cuts in a weighted directed graph

A simple algorithm is described for constructing a maximum packing of cuts directed away from a distinguished vertex, called the root, in a directed graph, each of whose edges has a nonnegative weight, and it is shown that the maximum packing value is equal to the weight of a minimum-weight spanning arborescence directed away from the root.     

On the distance matrix of a directed graph

In this note, we show how the determinant of the distance matrix D(G) of a weighted, directed graph G can be explicitly expressed in terms of the corresponding determinants for the (strong) blocks G_i of G. In particular, when cof D(G), the sum of the cofactors of D(G), does not vanish, we have the very attractive formula .     

On the spectral radius of a directed graph

We provide upper estimates on the spectral radius of a directed graph. In particular we prove that the spectral radius is bounded by the maximum of the geometric mean of in-degree and out-degree taken over all vertices. © 1996 John Wiley & Sons, Inc.     

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.     

Partition of a directed bipartite graph into two directed cycles

Let D = (V₁, V₂; A) be a directed bipartite graph with |V₁| = |V₂| = n 2. Suppose that d_D(x) + d_D(y) 3n + 1 for all x ε V₁ and y ε V₂. Then D contains two vertex-disjoint directed cycles of lengths 2n₁ and 2n₂, respectively, for any positive integer partition n = n₁ + n₂. Moreover, the condition is sharp for even n and nearly sharp for odd n.