Forbidden Subgraphs and Weak Locally Connected Graphs

Abstract

A graph is called H -free if it has no induced subgraph isomorphic to H. A graph is called $N^i$-locally connected if $G[\{ x\in V(G): 1\le d_G(w, x)\le i\}]$ is connected and $N_2$-locally connected if $G[\{uv: \{uw, vw\}\subseteq E(G)\}]$ is connected for every vertex w of G, respectively. In this paper, we prove the following.

• Every 2-connected $P_7$-free graph of minimum degree at least three other than the Petersen graph has a spanning Eulerian subgraph. This implies that every H-free 3-connected graph (or connected $N^4$-locally connected graph of minimum degree at least three) other than the Petersen graph is supereulerian if and only if H is an induced subgraph of $P_7$, where $P_i$ is a path of i vertices.

• Every 2-edge-connected H-free graph other than $\{K_{2, 2k+1}:k ~\text {is a positive integer}\}$ is supereulerian if and only if H is an induced subgraph of $P_4$.

• If every connected H-free $N^3$-locally connected graph other than the Petersen graph of minimum degree at least three is supereulerian, then H is an induced subgraph of $P_7$ or $T_{2, 2, 3}$, i.e., the graph obtained by identifying exactly one end vertex of $P_3, P_3, P_4$, respectively.

• If every 3-connected H-free $N_2$-locally connected graph is hamiltonian, then H is an induced subgraph of $K_{1,4}$.

We present an algorithm to find a collapsible subgraph of a graph with girth 4 whose idea is used to prove our first conclusion above. Finally, we propose that the reverse of the last two items would be true.