Related papers: An 8-flow theorem for signed graphs

Building on recently established enumerative connections between lambda calculus and the theory of embedded graphs (or "maps"), this paper develops an analogy between typing (of lambda terms) and coloring (of maps). Our starting point is…

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs,…

Converting modulo flows into integer-valued flows is one of the most critical steps in the study of integer flows. Tutte and Jaeger's pioneering work shows the equivalence of modulo flows and integer-valued flows for ordinary graphs.…

Given a $t$-$(v, k, \lambda)$ design, $\mathcal{D}=(X,\mathcal{B})$, a zero-sum $n$-flow of $\mathcal{D}$ is a map $f : \mathcal{B}\longrightarrow \{\pm1,\ldots, \pm(n-1)\}$ such that for any point $x\in X$, the sum of $f$ over all blocks…

A normal edge-coloring of a cubic graph is a proper edge-coloring, in which every edge is adjacent to edges colored with four distinct colors or to edges colored with two distinct colors. It is conjectured that $5$ colors suffice for a…

Let $S,T$ be two distinct finite Abelian groups with $|S|=|T|$. A fundamental theorem of Tutte shows that a graph admits a nowhere-zero $S$-flow if and only if it admits a nowhere-zero $T$-flow. Jaeger, Linial, Payan and Tarsi in 1992…

An indecomposable flow $f$ on a signed graph $\Sigma$ is a nontrivial integral flow that cannot be decomposed into $f=f_1+f_2$, where $f_1,f_2$ are nontrivial integral flows having the same sign (both $\geq 0$ or both $\leq 0$) at each edge…

This paper concerns a generalization of nowhere-zero modular q-flows from graphs to simplicial complexes of dimension d greater than 1. A modular q-flow of a simplicial complex is an element of the kernel of the d-th boundary map with…

A graph $G$ is $(d_1,\ldots,d_k)$-colorable if its vertex set can be partitioned into $k$ sets $V_1,\ldots,V_k$, such that for each $i\in\{1, \ldots, k\}$, the subgraph of $G$ induced by $V_i$ has maximum degree at most $d_i$. The Four…

The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree $\Delta\geq 3$ that is not the complete graph $K_{\Delta+1}$ or the complete bipartite graph $K_{\Delta,\Delta}$ admits an equitable coloring with $\Delta$…

Given a zero-sum function $\beta : V(G) \rightarrow \mathbb{Z}_3$ with $\sum_{v\in V(G)}\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \beta(v)$ in $\mathbb{Z}_3$ for every vertex $v\in V(G)$ is called a…

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at…

A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, Erd\H{o}s and Gy\'{a}rf\'{a}s conjectured that every graph of minimum degree at least three contains a cycle whose length is a…

Let $H$ and $G$ be graphs. An $H$-colouring of $G$ is a proper edge-colouring $f:E(G)\rightarrow E(H)$ such that for any vertex $u\in V(G)$ there exists a vertex $v\in V(H)$ with $f\left (\partial_Gu\right )=\partial_Hv$, where…

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each $k \in…

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a…

Hadwiger Conjecture has been an open problem for over a half century1,6, which says that there is at most a complete graph Kt but no Kt+1 for every t-colorable graph. A few cases of Hadwiger Conjecture, such as 1, 2, 3, 4, 5, 6-colorable…

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…

The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of…

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut…