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We consider the pattern detection problem in graphs: given a constant sizepattern graph $H$ and a host graph $G$, determine whether $G$ contains asubgraph isomorphic to $H$. Our main results are:  * We prove that if a pattern $H$ contains a $k$-clique subgraph, thendetecting whether an $n$ node host graph contains a not necessarily inducedcopy of $H$ requires at least the time for detecting whether an $n$ node graphcontains a $k$-clique. The previous result of this nature required that $H$contains a $k$-clique which is disjoint from all other $k$-cliques of $H$.  * We show that if the famous Hadwiger conjecture from graph theory is true,then detecting whether an $n$ node host graph contains a not necessarilyinduced copy of a pattern with chromatic number $t$ requires at least the timefor detecting whether an $n$ node graph contains a $t$-clique. This impliesthat: (1) under Hadwiger's conjecture for every $k$-node pattern $H$, findingan induced copy of $H$ requires at least the time of $\sqrt k$-cliquedetection, and at least size $\omega(n^{\sqrt{k}/4})$ for any constant depthcircuit, and (2) unconditionally, detecting an induced copy of a random$G(k,p)$ pattern w.h.p. requires at least the time of $\Theta(k/\log k)$-cliquedetection, and hence also at least size $n^{\Omega(k/\log k)}$ for circuits ofconstant depth.  * Finally, we consider the case when the pattern is a directed cycle on $k$nodes, and we would like to detect whether a directed $m$-edge graph $G$contains a $k$-Cycle as a not necessarily induced subgraph. We resolve a 14year old conjecture of [Yuster-Zwick SODA'04] on the complexity of $k$-Cycledetection by giving a tight analysis of their $k$-Cycle algorithm. Our analysisimproves the best bounds for $k$-Cycle detection in directed graphs, for all$k>5$.

Graph pattern detection: Hardness for all induced patterns and faster non-induced cycles