Flows on Signed Graphs

This dissertation focuses on integer flow problems within specific signed graphs. The theory of integer flows, which serves as a dual problem to vertex coloring of planar graphs, was initially introduced by Tutte as a tool related to the Four-Color Theorem. This theory has been extended to signed graphs. In 1983, Bouchet proposed a conjecture asserting that every flow-admissible signed graph admits a nowhere-zero 6-flow. To narrow dawn the focus, we investigate cubic signed graphs in Chapter 2. We prove that every flow-admissible 3-edge-colorable cubic signed graph admits a nowhere-zero 10-flow. This together with the 4-color theorem implies that every flow-admissible bridgeless planar signed graph admits a nowhere-zero 10-flow. As a byproduct of this research, we also demonstrate that every flow-admissible hamiltonian signed graph can admit a nowhere-zero 8-flow. In Chapter 3, we delve into triangularly connected signed graphs. Here, A triangle-path in a graph G is defined as a sequence of distinct triangles $T_1,T_2,\ldots,T_m$ in G such that for any i, j with $1\leq i \u3c j \leq m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j \u3e i+1$. We categorize a connected graph $G$ as triangularly connected if it can be demonstrated that for any two nonparallel edges $e$ and $e\u27$, there exists a triangle-path $T_1T_2\cdots T_m$ such that $e\in E(T_1)$ and $e\u27\in E(T_m)$. For ordinary graphs, Fan {\it et al.} characterized all triangularly connected graphs that admit nowhere-zero $3$-flows or $4$-flows. Corollaries of this result extended to integer flow in certain families of ordinary graphs, such as locally connected graphs due to Lai and certain types of products of graphs due to Imrich et al. In this dissertation, we extend Fan\u27s result for triangularly connected graphs to signed graphs. We proved that a flow-admissible triangularly connected signed graph $(G,\sigma)$ admits a nowhere-zero $4$-flow if and only if $(G,\sigma)$ is not the wheel $W_5$ associated with a specific signature. Moreover, this result is proven to be sharp since we identify infinitely many unbalanced triangularly connected signed graphs that can admit a nowhere-zero 4-flow but not 3-flow.\\ Chapter 4 investigates integer flow problems within $K_4$-minor free signed graphs. A minor of a graph $G$ refers to any graph that can be derived from $G$ through a series of vertex and edge deletions and edge contractions. A graph is considered $K_4$-minor free if $K_4$ is not a minor of $G$. While Bouchet\u27s conjecture is known to be tight for some signed graphs with a flow number of 6. Kompi\v{s}ov\\u27{a} and M\\u27{a}\v{c}ajov\\u27{a} extended those signed graph with a specific signature to a family \M, and they also put forward a conjecture that suggests if a flow-admissible signed graph does not admit a nowhere-zero 5-flow, then it belongs to \M. In this dissertation, we delve into the members in \M that are $K_4$-minor free, designating this subfamily as $\N$. We provide a proof demonstrating that every flow-admissible, $K_4$-minor free signed graph admits a nowhere-zero 5-flow if and only if it does not belong to the specified family $\N$

On flows of graphs

Tutte\u27s 3-flow Conjecture, 4-flow Conjecture, and 5-flow Conjecture are among the most fascinating problems in graph theory. In this dissertation, we mainly focus on the nowhere-zero integer flow of graphs, the circular flow of graphs and the bidirected flow of graphs. We confirm Tutte\u27s 3-flow Conjecture for the family of squares of graphs and the family of triangularly connected graphs. In fact, we obtain much stronger results on this conjecture in terms of group connectivity and get the complete characterization of such graphs in those families which do not admit nowhere-zero 3-flows. For the circular flows of graphs, we establish some sufficient conditions for a graph to have circular flow index less than 4, which generalizes a new known result to a large family of graphs. For the Bidirected Flow Conjecture, we prove it to be true for 6-edge connected graphs

Realizing degree sequences with graphs having nowhere-zero 3-flows

The following open problem was proposed by Archdeacon: Characterize all graphical sequences π such that some realization of π admits a nowhere-zero 3-flow. The purpose of this paper is to resolve this problem and present a complete characterization: A graphical sequence π = (d I,d2,...,dn) with minimum degree at least two has a realization that admits a nowhere-zero 3-flow if and only if π ≠ (34,2), (k,3k), (k2,3k-1), where k is an odd integer. © 2008 Society for Industrial and Applied Mathematics.published_or_final_versio

Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it. We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games.link_to_subscribed_fulltex

Realizing Degree Sequences with Graphs Having Nowhere-Zero 3-Flows

The following open problem was proposed by Archdeacon: Characterize all graphical sequences π such that some realization of π admits a nowhere-zero 3-flow. The purpose of this paper is to resolve this problem and present a complete characterization: A graphical sequence π = (d1, d2, ., dn) with minimum degree at least two has a realization that admits a nowhere-zero 3-flow if and only if π ≠ (34, 2), (k, 3k), (k2, 3k―1), where k is an odd integer

Graph Coloring Problems and Group Connectivity

1. Group connectivity. Let A be an abelian group and let iA(G) be the smallest positive integer m such that Lm(G) is A-connected. A path P of G is a normal divalent path if all internal vertices of P are of degree 2 in G and if |E(P)|= 2, then P is not in a 3-cycle of G. Let l(G) = max{lcub}m : G has a normal divalent path of length m{rcub}. We obtain the following result. (i) If |A| ≥ 4, then iA( G) ≤ l(G). (ii) If | A| ≥ 4, then iA(G) ≤ |V(G)| -- Delta(G). (iii) Suppose that |A| ≥ 4 and d = diam( G). If d ≤ |A| -- 1, then iA(G) ≤ d; and if d ≥ |A|, then iA(G) ≤ 2d -- |A| + 1. (iv) iZ 3 (G) ≤ l(G) + 2. All those bounds are best possible.;2. Modulo orientation. A mod (2p + 1)-orientation D is an orientation of G such that d +D(v) = d--D(v) (mod 2p + 1) for any vertex v ∈ V ( G). We prove that for any integer t ≥ 2, there exists a finite family F = F(p, t) of graphs that do not have a mod (2p + 1)-orientation, such that every graph G with independence number at most t either admits a mod (2p+1)-orientation or is contractible to a member in F. In particular, the graph family F(p, 2) is determined, and our results imply that every 8-edge-connected graph G with independence number at most two admits a mod 5-orientation.;3. Neighbor sum distinguishing total coloring. A proper total k-coloring &phis; of a graph G is a mapping from V(G) ∪ E(G) to {lcub}1,2, . . .,k{rcub} such that no adjacent or incident elements in V(G) ∪ E( G) receive the same color. Let m&phis;( v) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if m &phis;(u) ≠ m&phis;( v) for each edge uv ∈ E( G ). Let chitSigma(G) be the neighbor sum distinguishing total chromatic number of a graph G. Pilsniak and Wozniak conjectured that for any graph G, chitSigma( G) ≤ Delta(G) + 3. We show that if G is a graph with treewidth ℓ ≥ 3 and Delta(G) ≥ 2ℓ + 3, then chitSigma( G) + ℓ -- 1. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when ℓ = 3 and Delta ≥ 9, we show that Delta(G)+1 ≤ chit Sigma(G) ≤ Delta(G)+2 and characterize graphs with equalities.;4. Star edge coloring. A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Let ch\u27st(G) be the list star chromatic index of G: the minimum s such that for every s-list assignment L for the edges, G has a star edge coloring from L. By introducing a stronger coloring, we show with a very concise proof that the upper bound of the star chromatic index of trees also holds for list star chromatic index of trees, i.e. ch\u27st( T) ≤ [3Delta/2] for any tree T with maximum degree Delta. And then by applying some orientation technique we present two upper bounds for list star chromatic index of k-degenerate graphs