Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

We study the flow spectrum S(G) and the integer flow spectrum S(G) of odd regular graphs. We show that there are signed graphs where the difference between the integer flow number and the flow number is greater than or equal to 1, disproving a conjecture of Raspaud and Zhu [7]. Let G be a (2t + 1)-regular graph. We show that if r ∈ S(G), then r = 2 + 1t or r ≥ 2 + 2 2t−1 . This result generaliz...

A balanced graph is a bipartite graph with no induced circuit of length 2 (mod 4). These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for ...

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Hägglund constructed two graphs Blowup(K4, C) and Blowup(Prism,C4). Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs M0,1,2,...,k−2,k−1 which is obtained from cyclically 4-edge-connected and having a...

The bipartite density of a graphG is max{|E(H)|/|E(G)| : H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph. A biased maximum bipartite subgraph of a graph G is a bipartite subgraph of G with the maximum number of edges such that one of its partite sets is independent in G. Let H denote the collection of all connected cubic graphs ...

A 3-graph is a cubic graph endowed with a proper edge colouring in three colours. Surfaces can be modelled by means of 3-graphs. We show how 3-graphs can be used to establish the standard classification of sUrfaces by orientability and Euler characteristic. In [8], Tutte approaches topological graph theory from a combinatorial viewpoint. In particular, an entirely combinatorial approach to the ...

Every 2-connected simple cubic graph of order n ≥ 6 has a cycle cover with at most dn4 e cycles. This bound is best possible.