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A bridgeless graph $G$ is called $3$-flow-critical if it does not admit a nowhere-zero $3$-flow, but $G/e$ has for any $ein E(G)$. Tuttes $3$-flow conjecture can be equivalently stated as that every $3$-flow-critical graph contains a vertex of degree three. In this paper, we study the structure and extreme edge density of $3$-flow-critical graphs. We apply structure properties to obtain lower and upper bounds on the density of $3$-flow-critical graphs, that is, for any $3$-flow-critical graph $G$ on $n$ vertices, $$frac{8n-2}{5}le |E(G)|le 4n-10,$$ where each equality holds if and only if $G$ is $K_4$. We conjecture that every $3$-flow-critical graph on $nge 7$ vertices has at most $3n-8$ edges, which would be tight if true. For planar graphs, the best possible density upper bound of $3$-flow-critical graphs on $n$ vertices is $frac{5n-8}{2}$, known from a result of Kostochka and Yancey (JCTB 2014) on vertex coloring $4$-critical graphs by duality.
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A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs, Habilitationsschrift, Tu Ilmenau(1998)]. Voigt conjectured that if $G$ is a bipartite $3$-choice critical graph, then $G$ is $(4m, 2m)$-choosable for every integer $m$. This conjecture was disproved by Meng, Puleo and Zhu in [On (4, 2)-Choosable Graphs, Journal of Graph Theory 85(2):412-428(2017)]. They showed that if $G=Theta_{r,s,t}$ where $r,s,t$ have the same parity and $min{r,s,t} ge 3$, or $G=Theta_{2,2,2,2p}$ with $p ge 2$, then $G$ is bipartite $3$-choice critical, but not $(4,2)$-choosable. On the other hand, all the other bipartite 3-choice critical graphs are $(4,2)$-choosable. This paper strengthens the result of Meng, Puleo and Zhu and shows that all the other bipartite $3$-choice critical graphs are $(4m,2m)$-choosable for every integer $m$.
In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo $3$) between the inflow and outflow. They conjectured that all $5$-edge-connected graphs with a valid prescription function have a nowhere zero $3$-flow meeting that prescription. Kochol (2001) showed that replacing $4$-edge-connected with $5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lovasz et al.(2013) showed that both conjectures hold if the edge connectivity condition is relaxed to $6$-edge-connected. Both problems are still open for $5$-edge-connected graphs. The $3$-Flow Conjecture was known to hold for planar graphs, as it is the dual of Grotzschs Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al.(2016) provided the first direct proof using flows of the Strong $3$-Flow Conjecture for planar graphs. We prove the Strong $3$-Flow Conjecture for projective planar graphs.
A graph $G$ is emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $ein E(G)$. Melnikov and Steinberg [L. S. Melnikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with $n$ vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if $G$ is such a graph with $n(geq6)$ vertices, then $|E(G)|leq frac{5}{2}n-6 $, which improves the upper bound $frac{8}{3}n-frac{17}{3}$ given by Matsumoto [N. Matsumoto, The size of edge-critical uniquely 3-colorable planar graphs, Electron. J. Combin. 20 (3) (2013) $#$P49]. Furthermore, we find some edge-critical 3-colorable planar graphs which have $n(=10,12, 14)$ vertices and $frac{5}{2}n-7$ edges.
A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A graph $G$ is strongly fractional $r$-choosable if $G$ is $(a,b)$-choosable for all positive integers $a,b$ for which $a/b ge r$. The strong fractional choice number of $G$ is $ch_f^s(G) = inf {r: G $ is strongly fractional $r$-choosable$}$. This paper determines the strong fractional choice number of all $3$-choice critical graphs.
A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove that a connected graph $G$ is near-factor-critical if and only if it has a perfect matching. We also characterize disconnected near-factor-critical graphs.
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