Gallai's conjecture and the path number of odd semi-cliques

Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths including all the edges of G. Let $p\left(G\right)$ denote the minimum number of paths in a path decomposition of G. Gallai's Conjecture asserts that if G is connected, then $p\left(G\right)\le \lceil \frac{n}{2}\rceil$. The E-subgraph of G is the subgraph induced by the vertices of even degree in G. A well-known result of Lovász is that if the E-subgraph of G is empty or isomorphic to $K_1$, then $p\left(G\right)\le \lfloor \frac{n}{2}\rfloor$. In this paper, we prove that if the E-subgraph of G is isomorphic to $K_m$ with $m\le 15$, then $p\left(G\right)\le \lfloor \frac{n}{2}\rfloor +1$, which implies, under the condition, that Gallai's Conjecture holds when n is odd. A simple graph G on n vertices is called a semi-clique if $\left|E\right(G\left)\right|>\lfloor \frac{n}{2}\rfloor (n-1)$. By the definition, if G is a semi-clique on n vertices, then n must be odd and $p\left(G\right)\ge \lceil \frac{n}{2}\rceil$. As a corollary of our main result, we obtain that if G is a semi-clique on n vertices, then $p\left(G\right)\le \frac{4n+6}{7}$.