Anal. Theory Appl., 34 (2018), pp. 187-198.

Published online: 2018-07

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A loopless graph on n vertices in which vertices are connected at least by a
and at most by b edges is called a (a,b,n)-graph. A (b,b,n)-graph is called (b,n)-graph
and is denoted by K^{b}_{n}(it is a complete graph), its complement by K^{b}_{n}. A non increasing
sequence π = (d1,···,dn) of nonnegative integers is said to be (a,b,n) graphic if it
is realizable by an (a,b,n)-graph. We say a simple graphic sequence π = (d_{1},···,d_{n}) is
potentially K_{4}−K_{2}∪K_{2}-graphic if it has a a realization containing an K_{4}−K_{2}∪K_{2} as a
subgraph where K_{4} is a complete graph on four vertices and K_{2}∪K_{2} is a set of independent
edges. In this paper, we find the smallest degree sum such that every n-term
graphical sequence contains K_{4}−K_{2}∪K_{2} as subgraph.

A loopless graph on n vertices in which vertices are connected at least by a
and at most by b edges is called a (a,b,n)-graph. A (b,b,n)-graph is called (b,n)-graph
and is denoted by K^{b}_{n}(it is a complete graph), its complement by K^{b}_{n}. A non increasing
sequence π = (d1,···,dn) of nonnegative integers is said to be (a,b,n) graphic if it
is realizable by an (a,b,n)-graph. We say a simple graphic sequence π = (d_{1},···,d_{n}) is
potentially K_{4}−K_{2}∪K_{2}-graphic if it has a a realization containing an K_{4}−K_{2}∪K_{2} as a
subgraph where K_{4} is a complete graph on four vertices and K_{2}∪K_{2} is a set of independent
edges. In this paper, we find the smallest degree sum such that every n-term
graphical sequence contains K_{4}−K_{2}∪K_{2} as subgraph.

*Analysis in Theory and Applications*.

*34*(2). 187-198. doi:10.4208/ata.2018.v34.n2.8