scholarly journals On Graphs Having no Flow Roots in the Interval $(1,2)$

10.37236/3841 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
F.M. Dong
Keyword(s):  
Bridgeless Graph ◽  
Flow Polynomial

For any graph $G$, let $W(G)$ be the set of vertices in $G$ of degrees larger than 3. We show that for any bridgeless graph $G$, if $W(G)$ is dominated by some component of $G - W(G)$, then $F(G,\lambda)$ has no roots in the interval (1,2), where $F(G,\lambda)$ is the flow polynomial of $G$. This result generalizes the known result that $F(G,\lambda)$ has no roots in (1,2) whenever $|W(G)| \leq 2$. We also give some constructions to generate graphs whose flow polynomials have no roots in $(1,2)$.

scholarly journals On Graphs whose Flow Polynomials have Real Roots Only

10.37236/7512 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Fengming Dong
Keyword(s):  
Real Root ◽  
Plane Graph ◽  
Real Roots ◽  
Bridgeless Graph ◽  
Flow Polynomial

Let $G=(V,E)$ be a bridgeless graph. In 2011 Kung and Royle showed that the flow polynomial $F(G,\lambda)$ of $G$ has integral roots only if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether every graph whose flow polynomial has real roots only is the dual of some chordal and plane graph. We conclude that the answer for this problem is positive if and only if $F(G,\lambda)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then $P(G,\lambda)$ has real roots only if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,\lambda)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.

The NL-flow polynomial

2020 ◽  
Author(s):  
Barbara Altenbokum ◽  
Winfried Hochstättler ◽  
Johanna Wiehe
Keyword(s):  
Flow Polynomial

scholarly journals On spanning subgraphs of a connected bridgeless graph and their application to DT-graphs

1974 ◽  
Vol 16 (1) ◽  
pp. 17-28 ◽  
Author(s):  
Herbert Fleischner
Keyword(s):  
Spanning Subgraphs ◽  
Bridgeless Graph

A Tutte Polynomial for Maps

2018 ◽  
Vol 27 (6) ◽  
pp. 913-945 ◽  
Author(s):  
ANDREW GOODALL ◽  
THOMAS KRAJEWSKI ◽  
GUUS REGTS ◽  
LLUÍS VENA
Keyword(s):  
Finite Group ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Polynomial Invariant ◽  
Flow Polynomial ◽  
Orientable Surfaces

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

scholarly journals On the maximum fraction of edges covered by t perfect matchings in a cubic bridgeless graph

2015 ◽  
Vol 338 (8) ◽  
pp. 1509-1514
Author(s):  
Louis Esperet ◽  
Giuseppe Mazzuoccolo
Keyword(s):  
Perfect Matchings ◽  
Bridgeless Graph

scholarly journals Signed Cycle Double Covers

10.37236/6760 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Lingsheng Shi ◽  
Zhang Zhang
Keyword(s):  
Double Cover ◽  
Negative Sign ◽  
Signed Graph ◽  
Weak Version ◽  
Cycle Cover ◽  
Double Covers ◽  
Bridgeless Graph

The cycle double cover conjecture states that every bridgeless graph has a collection of cycles which together cover every edge of the graph exactly twice. A signed graph is a graph with each edge assigned by a positive or a negative sign. In this article, we prove a weak version of this conjecture that is the existence of a signed cycle double cover for all bridgeless graphs. We also show the relationships of the signed cycle double cover and other famous conjectures such as the Tutte flow conjectures and the shortest cycle cover conjecture etc.

scholarly journals A bound on the number of perfect matchings in Klee-graphs

2013 ◽  
Vol Vol. 15 no. 1 (Combinatorics) ◽  
Author(s):  
Marek Cygan ◽  
Marcin Pilipczuk ◽  
Riste Škrekovski
Keyword(s):  
Perfect Matchings ◽  
Famous Conjecture ◽  
International Audience ◽  
Bridgeless Graph

Combinatorics International audience The famous conjecture of Lovász and Plummer, very recently proven by Esperet et al. (2011), asserts that every cubic bridgeless graph has exponentially many perfect matchings. In this paper we improve the bound of Esperet et al. for a specific subclass of cubic bridgeless graphs called the Klee-graphs. We show that every Klee-graph with n ≥8 vertices has at least 3 *2(n+12)/60 perfect matchings.

scholarly journals Planar diagrams for local invariants of graphs in surfaces

2020 ◽  
Vol 29 (01) ◽  
pp. 1950093
Author(s):  
Calvin McPhail-Snyder ◽  
Kyle A. Miller
Keyword(s):  
Sufficient Condition ◽  
Cubic Graphs ◽  
Quantum Topology ◽  
Ribbon Graphs ◽  
Spatial Graphs ◽  
Diagram Category ◽  
Local Invariants ◽  
Flow Polynomial ◽  
Local Topology ◽  
Planar Diagram

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.

Factoring Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graph Theory ◽  
Bipartite Graphs ◽  
Petersen Graph ◽  
Different Types ◽  
Bridgeless Graph ◽  
Hall's Theorem ◽  
Hall’S Theorem

This chapter focuses on Hall's Theorem, introduced by British mathematician Philip Hall, and its connection to graph theory. It first considers problems that ask whether some collection of objects can be matched in some way to another collection of objects, with particular emphasis on how different types of schedulings are possible using a graph. It then examines one popular version of Hall's work, a statement known as the Marriage Theorem, the occurrence of matchings in bipartite graphs, Tutte's Theorem, Petersen's Theorem, and the Petersen graph. Peter Christian Julius Petersen introduced the Petersen graph to show that a cubic bridgeless graph need not be 1-factorable. The chapter concludes with an analysis of 1-factorable graphs, the 1-Factorization Conjecture, and 2-factorable graphs.

scholarly journals Oriented diameter and rainbow connection number of a graph

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Xiaolong Huang ◽  
Hengzhe Li ◽  
Xueliang Li ◽  
Yuefang Sun
Keyword(s):  
Minimum Degree ◽  
Edge Coloring ◽  
Integer Number ◽  
Colored Graph ◽  
Connection Number ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
International Audience ◽  
Bridgeless Graph ◽  
Rainbow Path

Graph Theory International audience The oriented diameter of a bridgeless graph G is min diam(H) | H is a strang orientation of G. A path in an edge-colored graph G, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer number k for which there exists a k-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of rad(G) and η(G), where rad(G) is the radius of G and η(G) is the smallest integer number such that every edge of G is contained in a cycle of length at most η(G). We also obtain constant bounds of the oriented diameter and the rainbow connection number for a (bipartite) graph G in terms of the minimum degree of G.

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