Since you asked in the ChatterBox for a recursive version, I thought I'd take up the challenge. Here's my take on a recursive solution:

#!/usr/bin/perl
use strict;
use warnings;

my %net = (
        '1.2.3.3' => ['1.2.3.4'],
        '1.2.3.4' => ['1.2.3.3', '1.2.3.5', '1.2.3.7'],
        '1.2.3.5' => ['1.2.3.4', '1.2.3.6', '1.2.3.7'],
        '1.2.3.6' => ['1.2.3.5', '1.2.3.7'],
        '1.2.3.7' => ['1.2.3.6', '1.2.3.4', '1.2.3.5'],
);

my %visited=();
my @stack=();

print find_path('1.2.3.4', '1.2.3.6') || "no solution!";

sub find_path {
        my ($beg, $end) = @_;

        return $end if $beg eq $end;

        $visited{$beg}=1;

        for my $try (@{$net{$beg}}) {
                if (!defined($visited{$try})) {
                        push @stack, $beg;
                        my $t = find_path($try, $end);
                        return $beg.'=>'.$t if $t;
                        pop @stack;
                }
        }
        undef;
}
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