Alon–Tarsi Number and Modulo Alon–Tarsi Number of Signed Graphs

Abstract(#br)We extend the concept of the Alon–Tarsi number for unsigned graph to signed one. Moreover, we introduce the modulo Alon–Tarsi number for a prime number p . We show that both the Alon–Tarsi number and modulo Alon–Tarsi number of a signed planar graph $$(G,\sigma )$$ ( G , σ ) are at most 5, where the former result generalizes Zhu’s result for unsigned case and the latter one implies that $$(G,\sigma )$$ ( G , σ ) is $${\mathbb {Z}}_5$$ Z 5 -colorable

Some orientation theorems for restricted DP-colorings of graphs

We define signable, generalized signable, and $Z$-signable correspondence assignments on multigraphs, which generalize good correspondence assignments as introduced by Kaul and Mudrock. DP-colorings from these classes generalize signed colorings, signed $\mathbb{Z}_p$-colorings, and signed list colorings of signed graphs. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for DP-colorings from $Z$-signable correspondence assignments on multigraphs, and obtain three DP-coloring analogs of the Alon-Tarsi theorem as corollaries.Comment: 15 page

List-avoiding orientations

Given a graph $G$ with a set $F(v)$ of forbidden values at each $v \in V(G)$, an $F$-avoiding orientation of $G$ is an orientation in which $deg^+(v) \not \in F(v)$ for each vertex $v$. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F(v)| < \frac{1}{2} deg(v)$ for each $v \in V(G)$, then $G$ has an $F$-avoiding orientation, and they showed that this statement is true when $\frac{1}{2}$ is replaced by $\frac{1}{4}$. In this paper, we take a step toward this conjecture by proving that if $|F(v)| < \lfloor \frac{1}{3} deg(v) \rfloor$ for each vertex $v$, then $G$ has an $F$-avoiding orientation. Furthermore, we show that if the maximum degree of $G$ is subexponential in terms of the minimum degree, then this coefficient of $\frac{1}{3}$ can be increased to $\sqrt{2} - 1 - o(1) \approx 0.414$. Our main tool is a new sufficient condition for the existence of an $F$-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi

Integer Flows and Circuit Covers of Graphs and Signed Graphs

The work in Chapter 2 is motivated by Tutte and Jaeger\u27s pioneering work on converting modulo flows into integer-valued flows for ordinary graphs. For a signed graphs (G, sigma), we first prove that for each k ∈ {lcub}2, 3{rcub}, if (G, sigma) is (k -- 1)-edge-connected and contains an even number of negative edges when k = 2, then every modulo k-flow of (G, sigma) can be converted into an integer-valued ( k + 1)-ow with a larger or the same support. We also prove that if (G, sigma) is odd-(2p+1)-edge-connected, then (G, sigma) admits a modulo circular (2 + 1/ p)-flows if and only if it admits an integer-valued circular (2 + 1/p)-flows, which improves all previous result by Xu and Zhang (DM2005), Schubert and Steffen (EJC2015), and Zhu (JCTB2015).;Shortest circuit cover conjecture is one of the major open problems in graph theory. It states that every bridgeless graph G contains a set of circuits F such that each edge is contained in at least one member of F and the length of F is at most 7/5&par;E(G)&par;. This concept was recently generalized to signed graphs by Macajova et al. (JGT2015). In Chapter 3, we improve their upper bound from 11&par;E( G)&par; to 14/3 &par;E(G)&par;, and if G is 2-edgeconnected and has even negativeness, then it can be further reduced to 11/3 &par;E(G)&par;.;Tutte\u27s 3-flow conjecture has been studied by many graph theorists in the last several decades. As a new approach to this conjecture, DeVos and Thomassen considered the vectors as ow values and found that there is a close relation between vector S1-flows and integer 3-NZFs. Motivated by their observation, in Chapter 4, we prove that if a graph G admits a vector S1-flow with rank at most two, then G admits an integer 3-NZF.;The concept of even factors is highly related to the famous Four Color Theorem. We conclude this dissertation in Chapter 5 with an improvement of a recent result by Chen and Fan (JCTB2016) on the upperbound of even factors. We show that if a graph G contains an even factor, then it contains an even factor H with.;&par;E(H)&par; ≥ 4/7 (&par; E(G)&par;+1)+ 1/7 &par;V2 (G)&par;, where V2( G) is the set of vertices of degree two

Nowhere-Zero Flow Polynomials

In this article we introduce the flow polynomial of a digraph and use it to study nowhere-zero flows from a commutative algebraic perspective. Using Hilbert's Nullstellensatz, we establish a relation between nowhere-zero flows and dual flows. For planar graphs this gives a relation between nowhere-zero flows and flows of their planar duals. It also yields an appealing proof that every bridgeless triangulated graph has a nowhere-zero four-flow

Matchings, factors and cycles in graphs

A matching in a graph is a set of pairwise nonadjacent edges, a k-factor is a k-regular spanning subgraph, and a cycle is a closed path. This thesis has two parts. In Part I (by far the larger part) we study sufficient conditions for structures involving matchings, factors and cycles. The three main types of conditions involve: the minimum degree; the degree sum of pairs of nonadjacent vertices (Ore-type conditions); and the neighbourhoods of independent sets of vertices. We show that most of our theorems are best possible by giving appropriate extremal graphs. We study Ore-type conditions for a graph to have a Hamilton cycle or 2-factor containing a given matching or path-system, and for any matching and single vertex to be contained in a cycle. We give Ore-type and neighbourhood conditions for a matching L of l edges to be contained in a matching of k edges (l 2) containing a given set of edges. We also establish neighbourhood conditions for the existence of a cycle of length at least k. A list-edge-colouring of a graph is an assignment of a colour to each edge from its own list of colours. In Part II we study edge colourings of powers of cycles, and prove the List-Edge-Colouring Conjecture for squares of cycles of odd length