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An Eulerian graph has an Euler circuit: a closed trail traversing every edge exactly once. Here the graph is a finite undirected graph, possibly a multigraph with self-loops. Parallel edges retain distinct identities, and each self-loop contributes two to its vertex degree. Repeated vertices are allowed. This models inspecting every road segment once, even when a junction must be revisited.

For a nonempty set of edges, the Euler circuit criterion requires a connected graph only on the edge support of a graph: isolated vertices impose no travel requirement. A single vertex with a self-loop and two vertices joined by two parallel edges are both Eulerian graphs. Neither requires three vertices. With no edges and at least one vertex, we allow the zero-length trail at that vertex as an Euler circuit; the graph with no vertices has no such walk.

The motivating Seven Bridges of Königsberg problem asks for an Eulerian trail, which need not return to its starting vertex. In Leonhard Euler's Solutio problematis ad geometriam situs pertinentis by Leonhard Euler, §2 poses the bridge-crossing question and §§4–6 encode land regions by letters. The river's shape drops out: only the connections matter. The paper was written in 1735 and published in 1741, according to the Euler Archive; the journal volume carries the nominal year 1736.^[ref]

An Eulerian trail, also called an Eulerian path, is a trail in a graph traversing every edge exactly once. The name Eulerian path does not require a graph path: vertices may repeat. A Hamiltonian path instead visits every vertex exactly once, and may omit edges. An Eulerian trail may omit isolated vertices, so these are different coverage requirements.

For a finite undirected multigraph with at least one edge, allowing self-loops, an Eulerian trail exists exactly when the edge support of a graph is a connected graph and there are zero or two vertices of odd vertex degree. In the two-odd-vertex case they must be the two distinct endpoints; in the zero-odd-vertex case every Eulerian trail is an Euler circuit. Count a self-loop twice in the vertex degree but once in the trail, and count each parallel edge separately.^[ref]

The parity obstruction is local: each intermediate visit pairs an arriving edge with a departing edge. Only the distinct endpoints can have an unpaired incidence. The handshaking lemma, ${\sum }_{v\in V}\mathrm{deg}(v)=2|E|$, also shows that the number of vertices with odd vertex degree is even. Sufficiency in the two-endpoint case reduces to the Euler circuit criterion: add one extra edge between the odd vertices, construct an Euler circuit, rotate its record to start with the extra edge, then delete that edge. The remaining trail has the required endpoints.

For the Seven Bridges of Königsberg multigraph, the vertex degrees at $A,B,C,D$ are $(5,3,3,3)$, so no Eulerian trail exists. Remove the unique edge $AD$: the vertex degrees become $(4,3,3,2)$, and a six-edge Eulerian trail runs from $B$ to $C$. Label the parallel edges $p_1,p_2$ between $A,B$ and $q_1,q_2$ between $A,C$; label $BD$ by $r$ and $CD$ by $s$. A complete record is $B,p_1,A,q_1,C,s,D,r,B,p_2,A,q_2,C$. It revisits $A,B,C$ while using every remaining edge once.

Explicit reference to laws is not necessary for every good scientific explanation. Laws remain indispensable in many explanations, but their occurrence alone is not sufficient either. The issue turns partly on whether a law means a contingent regularity of nature, any supporting generalization, or even a mathematical necessity.

The deductive-nomological model offers a strong positive case. Laws and circumstances entail the phenomenon, and at least one law is essential to the derivation. For example, Newton's second law and an inverse-square gravitational force, with appropriate initial conditions, yield predictions for an idealized two-body orbit. A universal relation unifies cases that otherwise look like unrelated observations. Carl Hempel and Paul Oppenheim explicitly develop adequacy conditions for such deductive explanations; their 1948 paper should not be read as a finished treatment of every statistical explanation.^[ref]

However, derivability does not always give the right explanatory direction. In a simplified geometrical optics model, a vertical pole of height $h$ on level ground casts a shadow of length $\ell =h/\tan \theta$ at solar elevation $\theta$. Given $h$ and $\theta$, one derives $\ell$; given $\ell$ and $\theta$, one derives $h$. The same geometrical relations enter both derivations. Yet the shadow does not normally explain why the pole was built to that height. The reverse calculation is a way to measure its height. The difference concerns which features produce or constrain which others, rather than deductive validity alone.

Mechanistic explanation is another challenge to a compulsory law-based format. A clock stopped because debris blocked a particular wheel, thereby preventing the mechanism from advancing. Inspecting the obstruction, removing it and observing restored operation can answer the causal question with considerable precision. The explanation describes organization and a relevant difference-making event without stating a universal law. It is more informative about this failure than the observation that every component obeyed Newtonian mechanics.

A defender of the covering-law model can reply that the causal account relies implicitly on regularities: debris that offers no resistance would not stop the wheel. This is a fair warning against treating a bare chronology as an explanation. But it does not establish that explanatory understanding requires identifying exceptionless fundamental laws, or that replacing the account with a complete microscopic derivation would improve the answer. Stable, tested relations in the actual operating regime may be sufficient.

An explanation can also turn on a mathematical constraint. Suppose an engineer asks why a connected network with four odd-degree junctions cannot be inspected by a route using every segment exactly once. Represent junctions as vertices and segments as edges of an undirected graph. Every intermediate visit pairs an incoming and an outgoing edge, so only the two endpoints of an open trail can have odd vertex degree. Four such vertices therefore exclude an Eulerian trail. The impossibility persists if segment lengths, construction materials or walking speeds change while the connections remain fixed. This is the obstruction in the Seven Bridges of Königsberg.^[ref]

The physical application requires that the graph represent the allowed routes correctly. Once that is granted, the explanatory necessity comes from parity, rather than a contingent physical law. Calling every mathematical constraint a law would preserve the original claim only by greatly widening its meaning. A critic may restrict “scientific” to causal explanations, but that restriction needs justification when applied mathematics answers a concrete engineering question.

Thus laws are powerful explanatory resources, and good explanations normally support relevant what-if judgements. Those judgements can be grounded in laws, mechanisms or mathematical constraints. Requiring explicit reference to laws imposes an unnecessarily narrow format; requiring some disciplined account of dependence is much more defensible.