Graph family operations

In previous papers, Catlin introduced four functions, denoted , , , and , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-flows for a fixed integer k3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.     

The harmonic and geometric mean of vectors

New vector operations are defined that generalize the classical harmonic and geometric means of scalars. The definition of the harmonic mean involves an explicit vector formula, whereas the geometric mean of two vectors is defined by an iterative procedure. Two special cases are described in some detail, the case of 2-vectors which are equivalent to the complex numbers, and the case of 4-vectors which are a representation of the quaternions.     

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E) be a balanced bipartite graph of order 2n. The path-cover numberpc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S⊆V(G) is called a balanced set of G if |S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of every length from |S|+2pc(〈S〉) up to 2n through S.     

Six-flows on almost balanced signed graphs

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. By Seymour's 6-flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow-admissible signed cubic graph with two negative edges admits a nowhere-zero 7-flow, and admits a nowhere-zero 6-flow if its underlying graph either contains a bridge, or is 3-edge-colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph.     

Weak solutions for diffusion-convection equations

Weak solutions to diffusion-convection equations with less regular convective field and boundary data are shown to exist as limits of solutions to problems with mixed boundary conditions.     

An inequality for the group chromatic number of a graph

We give an inequality for the group chromatic number of a graph as an extension of Brooks’ Theorem. Moreover, we obtain a structural theorem for graphs satisfying the equality and discuss applications of the theorem.     

On Group Connectivity of Graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z ₃-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph is Z ₃-connected. These two conjectures are in general open and few results are known so far. A weaker version of Tutte’s conjecture states that every 4-edge-connected graph with each edge contained in a circuit of length at most 3 admits a nowhere-zero Z ₃-flow. Devos proposed a stronger version problem by asking if every such graph is Z ₃-connected. In this paper, we first answer this later question in negative and get an infinite family of such graphs which are not Z ₃-connected. Moreover, motivated by these graphs, we prove that every 6-edge-connected graph whose edge set is an edge disjoint union of circuits of length at most 3 is Z ₃-connected. It is a partial result to Jaeger’s Z ₃-connectivity conjecture. Received: May 23, 2006. Final version received: January 13, 2008     

Rook theory and cycle-counting permutation statistics

A statistic is found to combinatorially generate the cycle-counting q-hit numbers, defined algebraically by Haglund [Adv. in Appl. Math. 17 (1996) 408–459]. We then define the notion of a cycle-Mahonian pair of statistics (generalizing that of a Mahonian statistic), and show that our newly discovered statistic is part of such a pair. Finally, we note a second example of a cycle-Mahonian pair of statistics which leads us to define the stronger property of being a cycle-Euler–Mahonian pair.     

A note on class one graphs with maximum degree six

It is known that there are class two graphs with Δ=6 which can be embedded in a surface Σ with Euler characteristic χ(Σ)?0. However, it is unknown whether there are class two graphs on the projective plane or on the plane with Δ=6. In this paper, we prove that every graph with Δ=6 is class one if it can be embedded in a surface with Euler characteristic at least -3 and is C₃-free, or C₄-free or if it can be embedded in a surface with Euler characteristic at least -1 and is C₅-free. This generalizes Zhou's results in [G. Zhou, A note on graphs of class I, Discrete Math. 263 (2003) 339-345] on planar graphs.