Maximum Orbit Weights in the σ-game and Lit-only σ-game on Grids and Graphs

Let G be a graph in which each vertex can be in one of two states: on or off. In the σ-game, when you “push” a vertex v you change the state of all of its neighbors, while in the σ⁺-game you change the state of v as well. Given a starting configuration of on vertices, the object of both games is to reduce it, by a sequence of pushes, to the smallest possible number of on vertices. We show that any starting configuration in a graph with no isolated vertices can, by a sequence of pushes, be reduced to at most half on, and we characterize those graphs for which you cannot do better. The proofs use techniques from coding theory. In the lit-only versions of these two games, you can only push vertices which are on. We obtain some results on the minimum number of on vertices one can obtain in grid graphs in the regular and lit-only versions of both games.     

Flows, flow-pair covers and cycle double covers

In this paper, some earlier results by Fleischner [H. Fleischner, Bipartizing matchings and Sabidussi’s compatibility conjecture, Discrete Math. 244 (2002) 77-82] about edge-disjoint bipartizing matchings of a cubic graph with a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow parity-pair-cover of G if the union of their supports covers the entire graph; f₁ is an h-flow and f₂ is a k-flow, and . Then G admits a nowhere-zero 6-flow if and only if G admits a (4,3)-flow parity-pair-cover; and G admits a nowhere-zero 5-flow if G admits a (3,3)-flow parity-pair-cover. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow even-disjoint-pair-cover of G if the union of their supports covers the entire graph, f₁ is an h-flow and f₂ is a k-flow, and for each {i,j}={1,2}. Then G has a 5-cycle double cover if G admits a (4,4)-flow even-disjoint-pair-cover; and G admits a (3,3)-flow parity-pair-cover if G has an orientable 5-cycle double cover.     

On Group Choosability of Total Graphs

In this paper, we study the group and list group colorings of total graphs and present group coloring versions of the total and list total colorings conjectures. We establish the group coloring version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs without certain small cycles, outerplanar graphs and near outerplanar graphs with maximum degree at least 4. In addition, the group version of the list total coloring conjecture is established for forests, outerplanar graphs and graphs with maximum degree at most two.     

On Spanning Disjoint Paths in Line Graphs

Spanning connectivity of graphs has been intensively investigated in the study of interconnection networks (Hsu and Lin, Graph Theory and Interconnection Networks, 2009). For a graph G and an integer s > 0 and for ${u, v \in V(G)}$ with u ≠ v, an (s; u, v)-path-system of G is a subgraph H consisting of s internally disjoint (u,v)-paths. A graph G is spanning s-connected if for any ${u, v \in V(G)}$ with u ≠ v, G has a spanning (s; u, v)-path-system. The spanning connectivity κ*(G) of a graph G is the largest integer s such that G has a spanning (k; u, v)-path-system, for any integer k with 1 ≤ k ≤ s, and for any ${u, v \in V(G)}$ with u ≠ v. An edge counter-part of κ*(G), defined as the supereulerian width of a graph G, has been investigated in Chen et al. (Supereulerian graphs with width s and s-collapsible graphs, 2012). In Catlin and Lai (Graph Theory, Combinatorics, and Applications, vol. 1, pp. 207–222, 1991) proved that if a graph G has 2 edge-disjoint spanning trees, and if L(G) is the line graph of G, then κ*(L(G)) ≥ 2 if and only if κ(L(G)) ≥ 3. In this paper, we extend this result and prove that for any integer k ≥ 2, if G ₀, the core of G, has k edge-disjoint spanning trees, then κ*(L(G)) ≥ k if and only if κ(L(G)) ≥ max{3, k}.     

Nowhere-Zero 3-Flows of Graphs with Independence Number Two

In this paper, we characterize all graphs with independence number at most 2 that admit nowhere-zero 3-flows.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

A note on shortest cycle covers of cubic graphs

Let SCC₃(G) be the length of a shortest 3‐cycle cover of a bridgeless cubic graph G. It is proved in this note that if G contains no circuit of length 5 (an improvement of Jackson's (JCTB 1994) result: if G has girth at least 7) and if all 5‐circuits of G are disjoint (a new upper bound of SCC₃(G) for the special class of graphs).