Group Connectivity of Graphs

Tutte introduced the theory of nowhere-zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere-zero A-flow, for any Abelian group A with |A| ≥ k. In 1992 Jaeger et al. [16] extended nowhere-zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b: V (G) A with sumv∈ V(G ) b(v) = 0, there always exists a map ƒ: E(G) A - {lcub}0{rcub}, such that at each v ∈ V(G), e=vw isdirectedfrom vtow fe- e=uvi sdirectedfrom utov fe=b v in A, then G is A-connected. For a 2-edge-connected graph G, define Lambda g(G) = min{lcub}k: for any Abelian group A with |A| ≥ k, G is A-connected{rcub}.;Let G1 ⊗ G2 and G1 xG2 denote the strong and Cartesian product of two connected nontrivial graphs G1 and G2. We prove that Lambdag(G 1 ⊗ G2) ≤ 4, where equality holds if and only if both G1 and G 2 are trees and min{lcub}|V (G1)|, |V (G2)|{rcub}=2; Lambda g(G1 ⊗ G 2) ≤ 5, where equality holds if and only if both G 1 and G2 are trees and either G 1 ≅ K1, m and G2 ≅ K 1,n, for n, m ≥ 2 or min{lcub}|V (G1)|, | V (G2)|{rcub}=2. A similar result for the lexicographical product graphs is also obtained.;Let P denote a path in G, let beta G(P) be the minimum length of a circuit containing P, and let betai(G) be the maximum of betaG(P) over paths of length i in G. We show that Lambda g(G) ≤ betai( G) + 1 for any integer i \u3e 0 and for any 2-connected graph G. Partial solutions toward determining the graphs for which equality holds were obtained by Fan et al. in [J. Comb. Theory, Ser. B, 98(6) (2008), 1325-1336], among others. We completely determine all graphs G with Lambda g(G) = beta2(G) + 1.;Let Z3 denote the cyclic group of order 3. In [16], Jaeger et al. conjectured that every 5-edge-connected graph is Z3 -connected. We proved the following: (i) Every 5-edge-connected graph is Z3 -connected if and only if every 5-edge-connected line graph is Z3 -connected. (ii) Every 6-edge-connected triangular line graph is Z3 -connected. (iii) Every 7-edge-connected triangular claw-free graph is Z3 -connected. In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere-zero 3-flow

Group Colorability and Hamiltonian Properties of Graphs

The research of my dissertation was motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Matthews and Sumner that every 4-connected claw-free graph is hamiltonian. Towards the hamiltonian line graph problem, we proved that every 3-edge-connected, essentially 4-edge-connected graph G has a spanning eulerian subgraph, if for every pair of adjacent vertices u and v, dG(u) + dG(v) ≥ 9. A straight forward corollary is that every 4-connected, essentially 6-connected line graph with minimum degree at least 7 is hamiltonian.;We also investigate graphs G such that the line graph L(G) is hamiltonian connected when L( G) is 4-connected. Ryjacek and Vrana recently further conjectured that every 4-connected line graph is hamiltonian-connected. In 2001, Kriesell proved that every 4-connected line graph of a claw free graph is hamiltonian connected. Recently, Lai et al showed that every 4-connected line graph of a quasi claw free graph is hamiltonian connected, and that every 4-connected line graph of an almost claw free graph is hamiltonian connected. In 2009, Broersma and Vumer discovered the P3-dominating (P3D) graphs as a superfamily that properly contains all quasi claw free graphs, and in particular, all claw-free graphs. Here we prove that every 4-connected line graph of a P3D graph is hamiltonian connected, which extends several former results in this area.;R. Gould [15] asked what natural graph properties of G and H are sufficient to imply that the product of G and H is hamiltonian. We first investigate the sufficient and necessary conditions for G x H being hamiltonian or traceable when G is a hamiltonian graph and H is a tree. Then we further investigate sufficient and necessary conditions for G x H being hamiltonian connected, or edge-pancyclic, or pan-connected.;The problem of group colorings of graphs is also investigated in this dissertation. Group coloring was first introduced by Jeager et al. [21]. They introduced a concept of group connectivity as a generalization of nowhere-zero flows. They also introduced group coloring as a dual concept to group connectivity. Prior research on group chromatic number was restricted to simple graphs, and considered only Abelian groups in the definition of chi g(G). The behavior of group coloring for multigraphs is different to that of simple graphs. Thus we extend the definition of group coloring by considering general groups (both Abelian groups and non-Abelian groups), and investigate the properties of chig for multigraphs by proving an analogue to Brooks\u27 Theorem

Normal 6-edge-colorings of some bridgeless cubic graphs

In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors such that each edge of the graph is normal. We denote by $\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal $7$-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal $6$-edge-coloring. Finally, we show that any bridgeless cubic graph $G$ admits a $6$-edge-coloring such that at least $\frac{7}{9}\cdot |E|$ edges of $G$ are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with arXiv:1804.0944

Claw -free graphs and line graphs

The research of my dissertation is motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Tutte that every 4-edge-connected graph has a no-where-zero 3-flow. Towards the hamiltonian line graph problem, we proved that every 3-connected N2-locally connected claw-free graph is hamiltonian, which was conjectured by Ryjacek in 1990; that every 4-connected line graph of an almost claw free graph is hamiltonian connected, and that every triangularly connected claw-free graph G with |E( G)| ≥ 3 is vertex pancyclic. Towards the second conjecture, we proved that every line graph of a 4-edge-connected graph is Z 3-connected

Cycles in graph theory and matroids

A circuit is a connected 2-regular graph. A cycle is a graph such that the degree of each vertex is even. A graph G is Hamiltonian if it has a spanning circuit, and Hamiltonian-connected if for every pair of distinct vertices u, v ∈ V( G), G has a spanning (u, v)-path. A graph G is s-Hamiltonian if for any S ⊆ V (G) of order at most s, G -- S has a Hamiltonian-circuit, and s-Hamiltonian connected if for any S ⊆ V( G) of order at most s, G -- S is Hamiltonian-connected. In this dissertation, we investigated sufficient conditions for Hamiltonian and Hamiltonian related properties in a graph or in a line graph. In particular, we obtained sufficient conditions in terms of connectivity only for a line graph to be Hamiltonian, and sufficient conditions in terms of degree for a graph to be s-Hamiltonian and s-Hamiltonian connected.;A cycle C of G is a spanning eulerian subgraph of G if C is connected and spanning. A graph G is supereulerian if G contains a spanning eulerian subgraph. If G has vertices v1, v2, &cdots; ,vn, the sequence (d( v1),d(v2), &cdots; ,d(vn)) is called a degree sequence of G. A sequence d = ( d1,d2, &cdots; ,dn) is graphic if there is a simple graph G with degree sequence d. Furthermore, G is called a realization of d. A sequence d ∈ G is line-hamiltonian if d has a realization G such that L(G) is hamiltonian. In this dissertation, we obtained sufficient conditions for a graphic degree sequence to have a supereulerian realization or to be line hamiltonian.;In 1960, Erdos and Posa characterized the graphs G which do not have two edge-disjoint circuits. In this dissertation, we successfully extended the results to regular matroids and characterized the regular matroids which do not have two disjoint circuits

Integer flows and Modulo Orientations

Tutte\u27s 3-flow conjecture (1970\u27s) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. A graph G admits a nowhere-zero 3-flow if and only if G has an orientation such that the out-degree equals the in-degree modulo 3 for every vertex. In the 1980ies Jaeger suggested some related conjectures. The generalized conjecture to modulo k-orientations, called circular flow conjecture, says that, for every odd natural number k, every (2k-2)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex. And the weaker conjecture he made, known as the weak 3-flow conjecture where he suggests that the constant 4 is replaced by any larger constant.;The weak version of the circular flow conjecture and the weak 3-flow conjecture are verified by Thomassen (JCTB 2012) recently. He proved that, for every odd natural number k, every (2k 2 + k)-edge-connected graph has an orientation such that the out-degree equals the in-degree modulo k for every vertex and for k = 3 the edge-connectivity 8 suffices. Those proofs are refined in this paper to give the same conclusions for 9 k-edge-connected graphs and for 6-edge-connected graphs when k = 3 respectively