TORSION IN THE MAGNITUDE HOMOLOGY OF GRAPHS

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Problem: Reconstructed statement. Work in finite simple undirected graphs with the usual shortest-path metric and integral magnitude homology $\mathrm{MH}_{k,\ell}(-)$. Let $C_4$ have cyclic vertices $a,b,c,d$. Attach two copies of $C_3$ to $C_4$ by identifying one edge of each triangle with an edge of $C_4$. If the two chosen edges of $C_4$ are not opposite—i.e. they are adjacent; and, if same-edge attachment is allowed, possibly equal—then the resulting graph $G$ is diagonal:

$$\mathrm{MH}_{k,\ell}(G)=0\qquad (k\ne \ell).$$

This is the natural formalization because the paper’s Figure 5 compares the adjacent-side and opposite-side cases for two triangles on a square.

Result: The conjecture is true.

Use the following standard fact: if $(X;A,B)$ is a projecting decomposition, then Hepworth–Willerton’s Mayer–Vietoris sequence gives a split exact sequence

$$0\to \mathrm{MH}(A\cap B)\to \mathrm{MH}(A)\oplus \mathrm{MH}(B)\to \mathrm{MH}(X)\to 0.$$

Hence, if $A,B,A\cap B$ are diagonal, then $X$ is diagonal.

First, $K_2$, $C_3$, and $C_4$ are diagonal. For $K_2$ and $C_3$, every nondegenerate $k$-path has length $k$. Also $C_4=K_2\square K_2$, so diagonality follows from the magnitude-homology Künneth theorem.

Let the first triangle be attached along edge $ab$. Let $P=C_4\cup C_3$. The decomposition

$$(P; C_4, C_3)$$

is projecting: the intersection is the edge $ab$, and $C_4$ projects to $ab$ by sending $d\mapsto a$, $c\mapsto b$, and fixing $a,b$. Thus $P$ is diagonal.

Now attach the second triangle along an adjacent edge, say $bc$. The decomposition

$$(G; P, C_3)$$

is projecting: the intersection is $bc$, and $P$ projects to $bc$ by

$$a\mapsto b,\qquad d\mapsto c,\qquad x\mapsto b,$$

where $x$ is the apex of the first triangle, while $b,c$ are fixed. The distance identities are immediate:

$$d(a,c)=2=d(a,b)+d(b,c),\quad d(d,b)=2=d(d,c)+d(c,b),\quad d(x,c)=2=d(x,b)+d(b,c).$$

Therefore Mayer–Vietoris again implies $G$ is diagonal.

If the wording is read as allowing both triangles on the same edge, the conclusion still holds: the two triangles sharing one base edge form the book graph

$$K_2\star \overline{K_2},$$

a join, hence diagonal by Hepworth–Willerton’s join theorem. Then decompose the whole graph as $C_4\cup (K_2\star \overline{K_2})$ along the common edge; since $C_4$ projects to that edge, Mayer–Vietoris again gives diagonality.

Thus in every non-opposite case,

$$\boxed{\mathrm{MH}_{k,\ell}(G)=0\text{ for }k\ne \ell.}$$

Citation: Uses Hepworth–Willerton’s Künneth theorem, join theorem, and Mayer–Vietoris theorem for projecting decompositions; see Hepworth and Willerton, “Categorifying the magnitude of a graph,” Homology Homotopy Appl. 19 (2017). The conjecture appears in Sazdanovic–Summers, “Torsion in the Magnitude Homology of Graphs,” Conjecture 5.3.