# Graphs with many edges avoided by Hamiltonian cycles

Let $$G$$ be a $$3$$-connected Hamiltonian graph with at least one edge that belongs to each H-cycle of $$G$$. Some authors (e.g. in the link given here) call such an edge an a-edge and an edge that belongs to no H-cycle of $$G$$ a b-edge. Let $$a(G)$$ and $$b(G)$$ denote the number of a-edges and b-edges, respectively. Define $$\rho(G)=\dfrac{b(G)}{a(G)}$$.

• Are there good (or even sharp) upper bounds for $$\rho(G)$$ ? Maybe $$\rho(G)<1$$?

Such graphs can be constructed e.g. using certain non-Hamiltonian graphs like the so-called Tutte fragment $$T$$ or the “Petersen fragment” $$P$$, the latter obtained from the Petersen graph by adding a vertex on each of three consecutive edges and an edge leaving at each of those vertices. These edges are labelled by $$1,2,3$$. It is easy to see that $$T$$ does not admit a H-path between 1 and 3, whereas $$P$$ admits no H-path starting at 2, only two H-paths between 1 and 3 (one consists of the violet and the red edges, the other one of the violet and the black edges). $$T$$ has two a-edges (marked in violet) and no b-edges; $$P$$ has nine a-edges (violet) and three b-edges (light blue).
The simplest $$3$$-connected Hamiltonian graphs obtained from these are a prism over the big “triangle” of $$T$$ with $$a(G)=3$$ and $$b(G)=0$$ (note that this one is even planar) and likewise a prism over $$P$$, which has $$a(G)=16$$ and $$b(G)=5$$.

By taking a Hamiltonian graph and replacing certain cubic vertices by $$T$$ or $$P$$, we can impose restrictions on the H-cycles and obtain graphs with various values of $$a(G)$$ and $$b(G)$$.
If we start with the wheel $$W_{n}$$ and replace one vertex by a $$T$$ in such a way that the spike becomes an a-edge, the resulting graph 'keeps' only two of the H-cycles of $$W_n$$ and has $$\rho(G)=\dfrac{n-3}{n}$$. I wonder if this is best possible. So if generally $$\rho(G)<1$$ holds, that would be sharp.

What about the maximum if we consider only cubic graphs ?
If we start with the prism over $$K_3$$ and make one ‘vertical' edge a b-edge by replacing one vertex with a $$P$$, we get a graph with a unique$$^*$$ H-cycle and $$\rho(G)=\dfrac{5}{13}\approx.3846$$.
If we start with the dodecahedron and 'block' three well chosen edges (i.e. make them into b-edges) by replacing three vertices with $$P$$'s, we get a graph with a unique$$^*$$ H-cycle and $$\rho(G)=\dfrac{16}{41}\approx.3902$$, slightly larger.
Starting with a truncated icosahedron (soccerball), we can still do better. It depends on how many edges have to be blocked by using $$P$$'s to remain with a unique$$^*$$ H-cycle: if there are $$k$$ such edges and the resulting graph $$G$$ is unique$$^*$$ Hamiltonian, it will have $$\rho(G)=\dfrac{30+2k}{60+7k}$$, which is equal to $$\dfrac{16}{41}$$ for $$k=9$$ and gets bigger as $$k$$ decreases. I've checked that $$k=7$$, thus $$\rho(G)=\dfrac{44}{109}\approx.4037$$, is possible. (Note that in the drawing given in the link, only the green and black 1-factors yield a H-cycle.)

$$^*$$ Edit: "unique" is meant w.r.t. the H-cycles of the original graph, because inside the instances of $$P$$ there are of course several ways of making H-paths through them.

Of course I don't claim that this method yields the best possible results. So the question:

• Are there cubic, 3-connected graphs with $$\rho(G)$$ even bigger than that?

The first counterexample given by @joro gives rise to the following family $G_k$ of graphs with $\rho(G_k)=k$ : $G_k$ has $n=4k+3$ vertices numbered $0,1,\dots,n-1$ forming a H-cycle (imagine a regular n-gon embedded such that the vertex $2k+1$ is on the bottom), plus the ‘horizontal’ chords $(2j-1,4k+3-2j)$ for $j=1,...,k$ (call their set $B$) and the other chords $(j,2k+1+j)$ for $j=0,...,2k+1$. It is not hard to see that $(4k+2,0)$ is an a-edge. As the graph without this edge and without $B$ is bipartite, all edges of $B$ must be b-edges. It remains to check that for each other edge, there is a H-cycle that doesn’t contain it.

For the second question, as @Martin Tancer and @nvcleemp have pointed aout, $\rho(G)\le1/2$ for cubic graphs. The following construction comes arbitrarily close to $1/2$ :

For $n=2k$, start with a H-cycle $0,...,2k-1,0$ and add the ‘horizontal’ chords $(j,2k-j)$ for $j=1,...,k-1$ and the ‘vertical’ chord $(0,k)$. Replace the vertex $k$ with an instance of $P$ that blocks the vertical chord. Then it is easy to see (starting at the vertex $0$) that the horizontal chords cannot be part of a H-cycle. So this graph has $a(G)=2k+7$ (the original cycle plus 7 inner edges of $P$) and $b(G)=k+1$ (the horizontal chords plus 2 inner edges of $P$).

So both original questions are answered, thanks to your input. Remains the interesting question raised by joro what can be said about $\rho(G)$ for 4-regular graphs.

• You mean the chords are modulo n? Verified with program for small $k$. – joro Dec 27 '13 at 8:59
• Yes modulo n, but the way I've written it, you don't even need that. I don't have time right now to add a diagram for say k=2 or k=3, but you can easily draw one. For checking that the edges (i,i+1) for i=0,...,n-2 aren't a-edges, it suffices to note that there are 2 H-cycles (symmetric to each other w.r.t. the vertical axis) which alternate essentially between diagonals of the n-gon and edges of it. – Wolfgang Dec 27 '13 at 9:55
• OK. Here is a drawing for k=3: s27.postimg.org/wwx38t3w3/graph_k_3.png – joro Dec 27 '13 at 10:04

The computer found counterexamples to $\rho(G)<1$.

Despite verification, I am not sure this is correct.

$G_1$ on $7$ vertices, $G_2$ on $11$ vertices. $\rho(G_1)=1,\rho(G_2)=2$

$G_1$:

edges=[(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 5), (3, 6), (4, 6)]
A=[(0, 3)]
B=[(4, 6)]


$G_2$:

edges=[(0, 5), (0, 6), (0, 7), (1, 6), (1, 7), (1, 8), (2, 6), (2, 8), (2, 9), (3, 7), (3, 9), (3, 10), (4, 8), (4, 9), (4, 10), (5, 9), (5, 10), (6, 8), (7, 10)]
A=[(0, 5)]
B=[(6, 8), (7, 10)]


Plot of $G_1$: • Thank you. Why should this not be correct? There are interesting patterns in your 2 graphs, which might allow (at least worth a try) to restrict further searches for even higher $\rho(G)$'s: Put $V=V_3\cup V_4$ (vertices of degree 3 and 4). Did you notice that G1 and G2 are "almost bipartite", the vertices of $V_3$ essentially alternating in each H-cycle with those of $V_4$. At the exception of 03 (05 for G2), which is... an a-edge! And the only edges in $\langle V_4\rangle$ are the b-edges. H-cycle e.g. for G1: 0 4 2 6 1 5 3, for G2: 0 6 2 8 1 7 3 9 4 10 5. – Wolfgang Dec 21 '13 at 13:25
• @Wolfgang if you trust my program, there are no cubic counterexamples rho(G) > 1 on less than 20 vertices. – joro Dec 21 '13 at 13:38
• Also note that displaying your $G_1$ and $G_2$ like you would display bipartite graphs, it is almost immediately clear why the a-edges and b-edges are what they are. BTW do you have a feeling that $\rho(G)$ is bounded? – Wolfgang Dec 21 '13 at 13:39
• For cubic ones, I conjecture $rho(G)\le 1/2$. Have you found cubic graphs with bigger rho? – Wolfgang Dec 21 '13 at 13:41
• Isn't $\rho(G) \leq 1/2$ somewhat trivial? A $b$-edge in a cubic graph is incident to four $a$-edges. Each $a$-edge comes this way from at most two $b$-edges. – Martin Tancer Dec 21 '13 at 15:08