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This is a note on three graph parameters motivated by the Euler-Poincarecharacteristic for simplicial complex. We show those three graph parameters ofa given connected graph $G$ is greater than or equal to that of the completegraph with $\max(h(G),\chi(G))$ vertices. This will yield three differentsimultaneous upperbounds of both the hadwiger number and chromatic number bymeans of the number of particular types of induced subgraphs. Some applicationsto Hadwiger's Conjecture is also discussed.

A Note on Graph Characteristics and Hadwiger's Conjecture