2. Degree Sequeces The cocept of degrees i graphs has provided a framewor for the study of various structural properties of graphs ad has therefore attracted the attetio of may graph theorists. Here we deliberate o the various criteria for a o-decreasig sequece of o-egative itegers to be a degree sequece of some graph. 2.1 Degree Sequeces Let d i, 1 i, be the degrees of the vertices v i of a graph i ay order. The sequece [d i ] 1 is called the degree sequece of the graph. The o-egative sequece [d i ] 1 is called the degree sequece of the graph if it is the degree sequece of some graph, ad the graph is said to realise the sequece. The set of distict o-egative itegers occurrig i a degree sequece of a graph is called its degree set. A set of o-egative itegers is called a degree set if it is the degree set of some graph, ad the graph is said to realise the degree set. Two graphs with the same degree sequece are said to be degree equivalet. I the graph of Figure 2.1(a), the degree sequece is D = [1, 2, 3, 3, 3, 4] or D = [1 2 3 3 4] ad its degree set is {1, 2, 3, 4}, while the degree sequece of the graph i Figure 2.1(b) is [1, 1, 2, 3, 3] ad its degree set is {1, 2, 3}. Fig. 2.1 If the degree sequece is arraged as the o-decreasig positive sequece d 1 1, d 2 2,... d, (d 1 < d 2 <... < d ), the sequece 1, 2,..., is called the frequecy sequece of the graph.
38 Degree Sequeces The two ecessary coditios implied by Theorem 1.1 ad Theorem 1.12 are ot sufficiet to esure that a o-egative sequece is a degree sequece of a graph. To see this, cosider the sequece [1, 2, 3, 4,..., 4, 1, 1]. The sum of the degrees is clearly eve ad = 1. However, this is ot a degree sequece, sice there are two vertices with degree 1, ad this requires that each of the two vertices is joied to all the other vertices, ad therefore δ 2. But the miimum umber i the sequece is 1. A degree sequece is perfect if o two of its elemets are equal, that is, if the frequecy sequece is 1, 1,..., 1. A degree sequece is quasi-perfect if exactly two of its elemets are same. Defiitio: Let D = [d i ] 1 be a o-egative sequece ad be ay iteger 1. Let D = [d i ] 1 be the sequece obtaied from D by settig d = 0 ad d i = d i 1 for the d largest elemets of D other tha d. Let H be the graph obtaied o the vertex set V = {v 1, v 2,..., v } by joiig v to the d vertices correspodig to the d elemets used to obtai D. This operatio of gettig D ad H is called layig off d ad D is called the residual sequece, ad H the subgraph obtaied by layig off d. Example Let D = [2, 2, 3, 3, 4, 4]. Tae d 3 = 0. The D = [2, 2, 0, 2, 3, 3]. The subgraph H i this case is show i Figure 2.2. Fig. 2.2 2.2 Criteria for Degree Sequeces Havel [112] ad Haimi [99] idepedetly obtaied recursive ecessary ad sufficiet coditios for a degree sequece, i terms of layig off a largest iteger i the sequece. Wag ad Kleitma [261] proved the ecessary ad sufficiet coditios for arbitrary layoffs. Theorem 2.1 A o-egative sequece is a degree sequece if ad oly if the residual sequece obtaied by layig off ay o-zero elemet of the sequece is a degree sequece. Proof Sufficiecy Let the o-egative sequece be [d i ] 1. Suppose d is the o-zero elemet laid off ad the residual sequece [d i ] 1 is a degree sequece. The there exists a graph G
Graph Theory 39 realisig [d i ] 1 i which v has degree zero ad some d vertices, say v i j, 1 j d have degree d i j 1. Now, by joiig v to these vertices we get a graph G with degree sequece [d i ] 1. (Observe that the subgraph obtaied by such joiig is precisely the subgraph H obtaied by layig off d ). Necessity We are give that there is a graph realisig D = [d i ] 1. Let d be the elemet to be laid off. First, we claim there is a graph realisig D i which v is adjacet to all the vertices i the set S of d largest elemets of D {d }. If ot, let G be a graph realisig D such that v is adjacet to the maximum possible umber of vertices i S. The there is a vertex v i i S to which v is ot adjacet ad hece a vertex v j outside S to which v is adjacet (sice d(v ) = S ). By defiitio of S, d j d i. Therefore there is a vertex v h i V {v } adjacet to v i, but ot adjacet to v j. Note that v h may be i S (Fig. 2.3). Fig. 2.3 Costruct a graph H from G by deletig the edges v j v ad v h v i ad addig the edges v j v h ad v i v. This operatio does ot chage the degree sequece. Thus H is a graph realisig the give sequece, i which oe more vertex, amely v i of S is adjacet to v, tha i G. This cotradicts the choice of G ad establishes the claim. To complete the proof, if G is a graph realisig the give sequece ad i which v is adjacet to all vertices of S, let G = G v. The G has the residual degree sequece obtaied by layig off d. Defiitio: Let the subgraph H o the vertices v i, v j, v r, v s of a multigraph G cotai the edges v i v j ad v r v s. The operatio of deletig these edges ad itroducig a pair of ew edges v i v s ad v j v r, or v i v r ad v j v s is called a elemetary degree preservig trasformatio (EDT), or simple exchage, or 2-switchig, or elemetary degree-ivariat trasformatio. Remars 1. The result of a EDT is clearly a degree equivalet multigraph. 2. If a EDT is applied to a graph, the result will be a graph oly if the latter pair of edges (v i v s ad v j v r ), or (v i v r ad v j v s ) does ot exist i G.
40 Degree Sequeces Theorem 2.2 (Havel, Haimi) The o-egative iteger sequece D = [d i ] 1 is graphic if ad oly if D is graphic, where D is the sequece (havig 1 elemets) obtaied from D by deletig its largest elemet ad subtractig 1 from its ext largest elemets. Proof Sufficiecy Let D = [d i ] 1 be the o-egative sequece with d 1 d 2... d. Let G be the graph realisig the sequece D. We add a ew vertex adjacet to vertices i G havig degrees d 2 1,..., d +1 1. Those d i are the largest elemets of D after itself. (But the umbers d 2 1,..., d +1 1 eed ot be the largest elemets i D ). Necessity Let G be a graph realisig D = [d i ] 1, d 1 d 2... d. We produce a graph G realisig D, where D is the sequece obtaied from D by deletig the largest etry d 1 ad subtractig 1 from d 1 ext largest etries. Let w be a vertex of degree d 1 i G ad N(w) be the set of vertices which are adjacet to w. Let S be the set of d 1 umber of vertices i G havig the desired degrees d 2,..., d d1 +1. If N(w) = S, we ca delete w to obtai G. Otherwise, some vertex of S is missig from N(w). I this case, we modify G to icrease N(w) S without chagig the degree of ay vertex. Sice N(w) S ca icrease at most d 1 times, repeatig this procedure coverts a arbitrary graph G that realises D, ito a graph G that realises D, ad has N(w) = S. From G, we the delete w to obtai the desired graph G realisig D. If N(w) S, let x S ad z / S, so that wz is a edge ad wx is ot a edge, sice d(w) = d 1 = S. By this choice of S, d(x) d(z) (Fig. 2.4). Fig. 2.4 We would lie to add wx ad delete wz without chagig their respective degrees. It suffices to fid a vertex y outside T = {x, z, w} such that yx is a edge, while yz is ot. If such a y exists, the we also delete xy ad add zy. Let q be the umber of copies of the edge xz (0 or 1). Now x has d(x) q eighbours outside T, ad z has d(z) 1 q eighbours outside T. Sice d(x) d(z), the desired y outside T exists ad we ca perform the EDT (elemetary degree preservig trasformatio or 2-switch). Algorithm: The above recursive coditios give a algorithm to chec whether a oegative sequece is a degree sequece ad if so to costruct a graph realisig it.
Graph Theory 41 The algorithm starts with a empty graph o vertex set V = {v 1, v 2,..., v } ad at the th iteratio geerates a subgraph H of G by deletig (layig off) a vertex of maximum degree i the residual sequece at that stage. If the give sequece is a degree sequece, we ed up with a ull degree sequece (i.e., for each i, d i = 0) ad the graph realisig the origial sequece is simply the sum of the subgraphs H j. If ot, at some stage, oe of the elemets of the residual sequece becomes egative, ad the algorithm reports o-realisability of the sequece. A obvious modificatio of the algorithm, obtaied by choosig a arbitrary vertex of positive degree, gives the Wag-Kleitma algorithm for geeratig a graph with a give degree sequece. Remars 1. There ca be may o-isomorphic graphs with the same degree sequece. The smallest example is the pair show i Figure 2.5 o five vertices with the degree sequece [2, 2, 2, 1, 1]. Fig. 2.5 The problem of geeratig all o-isomorphic graphs of give order ad size ivolves the problem of graph isomorphism for which a good algorithm is ot yet ow. So also is the problem of geeratig all o-isomorphic graphs with give degree sequece. I fact, eve the problem of fidig the umber of o-isomorphic graphs with give order ad size, or with give degree sequece (ad several other problems of similar ature) has ot bee satisfactorily solved. 2. The Wag-Kleitma algorithm is certaily more geeral tha the Havel-Haimi algorithm, as it ca geerate more umber of o-isomorphic graphs with a give degree sequece, because of the arbitrariess of the laid-off vertex. For example, ot all the five o-isomorphic graphs with the degree sequece [3, 3, 2, 2, 1, 1] ca be geerated by the Havel-Haimi algorithm ulie the Wag-Kleitma algorithm. 3. Eve the Wag-Kleithma algorithm caot always geerate all graphs with a give degree sequece. For example, the graph G with degree sequece [3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1] show i Figure 2.6, caot be geerated by this algorithm. For a. if we lay off a 3, it has to be laid off agaist the other 3 s ad will geerate a graph i which a vertex with degree 3 is adjacet to three other vertices with degree 3, b. if we lay off a 2 it will geerate a graph with a vertex of degree 2 adjacet to two vertices of degree 3,
42 Degree Sequeces c. if we lay off a oe it will geerate a graph i which a vertex of degree oe is adjacet to a vertex of degree 3. Noe of these cases is realised i the give graph G. Fig. 2.6 However, there are other methods of geeratig all graphs realisig a degree sequece D from ay oe graph realisig D based o a theorem by Haimi [98]. But those will also be iefficiet uless some efficiet isomorphism testig is developed. 4. The graphs i Figure 2.5 show that the same degree sequece may be realised by a coected as well as a discoected graph. Such degree sequeces are called potetially coected, where as a degree sequece D such that every graph realisig D is coected is called a forcibly coected degree sequece. Defiitio: If P is a graph property, ad D = [d i ] 1 is a degree sequece, the D is said to be potetially-p, if at least oe graph realisig D is a P-graph, ad it is said to be forcibly-p if every graph realisig it is a P-graph. Theorem 2.3 (Haimi) If G 1 ad G 2 are degree equivalet graphs, the oe ca be obtaied from the other by a fiite sequece of EDTs. Proof Superimpose G 1 ad G 2 such that each vertex of G 2 coicides with a vertex of G 1 with the same degree. Imagie the edges of G 1 are coloured blue ad the edges of G 2 are coloured red. The i the superimposed multigraph H, the umber of blue edges icidet equals the umber of red edges icidet at every vertex. We refer to this as blue-red parity. If there is a blue edge v i v j ad a red edge v i v j i H, we call it a blue-red parallel pair. Let K be the graph obtaied from H by deletig all such parallel pairs. The K is the ull graph if ad oly if G 1 ad G 2 are label-isomorphic i H ad hece origially isomorphic. If this is ot the case, we show that we ca create more parallel pairs by a sequece of EDTs ad delete them till the fial resultat graph is ull. This will prove the theorem. Let B ad R deote the sets of blue ad red edges i K. If v i v j B, we show that we ca produce a parallel pair at v i v j, so that the pair ca be deleted. This would establish the claim made above. Now, by costructio, there is a blue-red degree parity at every vertex of K. So there are red edges v i v, v j v r i K. If v v r (Fig. 2.7(a)) a EDT i G 2 switchig the red edges to v i v j, v v r produces a blue-red parallel at v i v j.
Graph Theory 43 Fig. 2.7 If v = v r, agai by degree parity, at v there are at least two blue edges. Let v v s be oe such blue edge. The v s is distict from both v i ad v j, for otherwise, there is a blue-red parallel pair v i v or v j v r. The there is aother red edge v s v t, v t distict from v i or v j. Let v t v i. The two subcases v t = v j ad v t v j are show i Figure 2.7(b) ad (c). I the case of (b), oe EDT of G 2 switchig v i v ad v s v t to positios v i v j ad v s v produces a bluered pair at v i v j ad v v s. I the case of (c), oe EDT of G 2 switchig v i v ad v t v s to positios v s v ad v t v i produces a blue-red parallel pair at v v s (which ca be deleted). Aother EDT of G 2 switchig the blue-red pair v t v i ad v j v to positios v i v j ad v s v produces a blue-red pair v i v j. Sice i both cases we get a blue-red pair at v i v j positio, our claim is established ad the proof of the theorem is complete. Remars I the related cotext of a (0, 1) matrix A (that is, a matrix A whose elemets are 0 s or 1 s), Ryser [227] defied a ( iterchage ) as a trasformatio of( the elemets ) of 1 0 0 1 A that chages a mior of type A 1 = ito a mior of the type A 1 =, or vice 0 1 1 0 versa ad proved a iterchage theorem which ca be iterpreted as EDT theorem for bipartite graphs ad digraphs. The ext result is a combiatorial characterisatio of degree sequeces, due to Erdos ad Gallai [73]. Several proofs of the criterio exist; the first proof give here is due to Choudam [58] ad the secod oe is due to Tripathi et al [246]. Theorem 2.4 (Erdos-Gallai) A o-icreasig sequece [d i ] 1 of o-egative itegers is a degree sequece if ad oly if D = [d i ] 1 is eve ad the iequality
44 Degree Sequeces d i ( 1)+ i=+1 is satisfied for each iteger, 1. Proof Necessity mi(d i, ) (2.4.1) Evidetly d i is eve. Let U deote the subset of vertices with the highest degrees i D. The the sum s = d i ca be split as s 1 + s 2, where s 1 is the cotributio to s from edges joiig vertices i U, each edge cotributig 2 to the sum, ad s 2 is the cotributio to s from the edges betwee vertices i U ad U (where U = V U), each edge cotributig 1 to the sum (Fig. 2.8). s 1 is clearly bouded above by the degree sum of a complete graph o -vertices, i.e., ( 1). Also, each vertex v i of U ca be joied to at most mi (d i, ) vertices of U, so that s 2 is bouded above by mi(d i, ). Together, we get (2.4.1). i=+1 Fig. 2.8 Sufficiecy We iduct o the sum s = d i ad use the obvious iequality mi(a, b) 1 mi(a 1, b), (2.4.2) for positive itegers a ad b. For s = 2, clearly K 2 ( 2)K 1 realises the oly sequece [1, 1, 0, 0,... 0] or [1 2 0 2 ] satisfyig the coditios (2.4.1). As iductio hypothesis, let all o-icreasig sequeces of o-egative itegers with eve sum at most s 2 ad satisfyig (2.4.1) be degree sequeces. Let D = [d i ] 1 be a sequece with sum s ad satisfyig (2.4.1). We produce a ew oicreasig sequece D of o-egative itegers by subtractig oe each from two positive terms of D ad verify that D satisfies the hypothesis of the theorem. Sice the trailig
Graph Theory 45 zeros i the o-icreasig sequeces of o-egative itegers do ot essetially affect the argumet, there is o loss of geerality i assumig that d > 0, ad we assume this to simplify the expressio. To defie D, let t be the smallest iteger ( 1) such that d t > d t+1. That is, let D be d 1 = d 2 =... = d t > d t+1 d t+2... d > 0. If D is regular (that is, d i = d > 0, for all i) the let t be 1. The d i = d i, f or 1 i t 1 ad t + 1 i 1, d t 1, f or i = t, d 1, f or i =. Clearly, D is a o-icreasig sequece of o-egative itegers ad d i = s 2 is eve. We verify that D satisfies (2.4.1) by cosiderig several cases depedig o the relative positio of ad the magitudes of d ad d. Case I for D. Let =. Therefore, Case II Let t 1. The d i = d i 1 ( 1)+ Therefore, d i = d i 2 ( 1) 2 < ( 1) = RHS of (2.4.1) i=+1 = ( 1)+ 1 mi(d i, )+mi(d, ) 1 i=+1 mi(d i, ) 1 (sice D satisfies (2.4.1)) ( 1)+ 1 mi(d i, )+mi(d 1, ) by (2.4.2) i=+1 = ( 1)+ 1 mi(d i, )+mi(d, ) i=+1 d i Case III Let t 1. ( 1)+ i=+1 Subcase III.1 Assume d 1. mi(d i, ). The d i = d ( 1) ( 1)+ sice the secod term is o-egative. i=+1 mi(d i, ), Subcase III.2 Every d j =, 1 j. We first observe that d +2 +...+ d 2.
46 Degree Sequeces This is obvious if +2 1, because d > 0 gives d 1 ad d 1 1. Whe +2 =, we have = 2. As t 1, t + 1 = 2+1 = 1. Sice t > 1 is ot possible, t = 1. The sequece D is [ 2, 2,..., 2, d ], or [( 2) 1 d ]. The s = ( 1) ( 2)+d. Sice s is eve, d is eve ad hece d 2. Thus, d +2 +...+ d 2. Therefore, d +2 +...+ d 2 0. Now, d i = d i =. = 2 = 2 + = 2 +d +1, (because t 1, ad d 1 =... = d t 1 = d t, so if d t 1 =, the d t =, ad if d =, d +1 = ). Thus, d i 2 +d +1 +(d +2 +...+ d 2) = ( 1)+ i=+1 mi(d i, ) 2, (because mi (d +1, ) = d +1, mi (d +2, ) = = d +2,..., mi (d t, ) = = d t,..., mi (d t+1, ) = d t+1 (as d t+1 < d t = ),..., mi (d, ) = d (as d < d t = )). Hece, d i ( 1)+ = ( 1)+ ( 1)+ i=+1 i t, i=+1 i t, i=+1 i t, mi(d i, )+mi(d t, )+mi(d, ) 2 mi(d i, )+mi(d t + 1, )+mi(d + 1, ) 2 mi(d i, )+mi(d t, )+1+mi(d, )+1 2 = ( 1)+ mi(d i, ). i=+1 Subcase III.3 Let d +1. i. Let d +1. The d i = d i ( 1)+ i=+1 mi(d i, ) (sice D satisfies (2.4.1)) = ( 1)+ mi(d i, )+mi(d t, )+mi(d, ) +1 i t,
Graph Theory 47 = ( 1)+ mi(d i, )+mi(d t 1, )+mi(d 1, ), +1 i t, (because mi(d t, ) = mi(d t 1, ) =, mi(d, ) = mi(d 1, ) =, as d t + 1, d +1 implies that d t 1, d 1 ). So, d i ( 1)+ mi(d i, )+mi(d t, )+mi(d, ) +1 i t, = ( 1)+ mi(d i, ). i=+1 ii. Let d ad let r be the smallest iteger such that d t+r+1. We verify that i (2.4.1), D ca ot attai equality for such a choice of. For, with equality, we have d i = d = ( 1)+ t+r +1 mi(d i, )+ t+r+1 mi(d i, ) = ( 1)+(t + r )+ d i, t+r+1, f or i = +1,..., t + r as d i +1, because mi (d i, ) = d i, f or i = t + r+ 1,..., as d i. So, d = (t + r 1)+ t+r+1 { The +1 d i = (+1)d = (+1) (t + r 1)+ 1 d i. = (+1)(t + r 1)+ +1 t+r+1 t+r+1 = (+1) (+ 1)+(+ 1)(t + r 1)+ = (+1)+(+ 1)(t + r 1)+ = (+1)+ t+r+1 mi (d i, +1), d i }, (usig d from above) d i > (+1)(t + r 1)+ t+r+1 d i t+r+1 t+r d i = (+1)+ d i t+r+1 +1 (+1)+ d i t+r+1
48 Degree Sequeces because mi (d i, +1) = +1 for i = +1,..., t + r, ad mi (d i, +1) = d i, for i = t + r+ 1,...,. So, +1 d i > (+1)+ +1 mi (d i, +1). Therefore, +1 d i > (+1)+(+1)+ +2 mi (d i, +1), which is a cotradictio to (2.4.1), for D for + 1. Hece D has strict iequality for. Therefore, Thus, d i = d i = d i < ( 1)+ d i ( 1)+ = ( 1)+ ( 1)+ 1 i=+1 i t 1 i=+1 +1 +1 mi (d i, ). mi (d i, ) 1 mi (d i, )+mi (d t, )+mi (d, ) 1 mi (d i, )+mi (d t 1, )+mi (d 1, ), i t as mi (d, ) 1 mi (d 1, ), mi(d t, ) = (sice d t + 1), mi(d t 1, ) = (sice d t 1 ). Therefore, d i ( 1)+ +1 mi (d i, ). Hece i all cases D satisfies (2.4.1). Therefore by iductio hypothesis, there is a graph G realisig D. If v t v / E(G ), the G + v t v gives a realisatio G of D. If v t v E(G ), sice d(v t G ) = d t 1 2, there is a vertex v r such that v r v t / E(G ). Also, sice d(v r G ) > d(v G ), there is a vertex v s such that v s v / E(G ). Maig a EDT exchagig the edge pair v t v, v r v s for the edge pair v t v r, v s v, we get a realisatio G of D with v t v / E(G ). The G + v t v realises D. Secod Proof of Sufficiecy (Tripathi et al.) Let a subrealisatio of a o-icreasig sequece [d 1, d 1,..., d ] be a graph with vertices v 1, v 1,..., v such that d(v i ) = d i for 1 i, where d(v i ) deotes the degree of v i. Give a sequece [d 1, d 1,..., d ] with a eve sum that satisfies (2.4.1), we costruct a realisatio through successive subrealisatios. The iitial subrealisatio has vertices ad o edges.
Graph Theory 49 I a subrealisatio, the critical idex r is the largest idex such that d(v i ) = d i for 1 i < r. Iitially, r = 1 uless the sequece is all 0, i which case the process is complete. While r, we obtai a ew subrealisatio with smaller deficiecy d r d(v r ) at vertex v r while ot chagig the degree of ay vertex v i with i < r (the degree sequece icreases lexicograpically). The process ca oly stop whe the subrealisatio of d. Let S = {v r+1,..., v }. We maitai the coditio that S is a idepedet set, which certaily holds iitially. Write u i v j whe v i v j E(G); otherwise, v i v j Case 0 v r v i for some vertex v i such that d(v i ) < d i. Add the edge u r v i. Case 1 v r v i for some i with i < r. Sice d(v i ) = d i d r > d(v r ), there exists u N(u i ) (N(v r ) {v r }), where N(z) = {y : z y}. If d r d(v r ) 2, the replace uv i with {uv r, v i v r }. If d r d(v r ) = 1, the sice d i d(v i ) is eve there is a idex with > r such that d(v ) < d. Case 0 applies uless v r v ; replace {v r v, uv i } with {uv r, v i u r }. Case 2 v 1,..., v r 1 N(v r ) ad d(v ) mi{r, d } for some with > r. I a subrealisatio, d(v ) d. Sice S is idepedet, d(v ) r. Hece d(v ) < mi{r, d }, ad case 0 applies uless u v r. Sice d(v ) < r, there exists i with i < r such that u v i. Sice d(v i ) > d(v r ), there exists u N(v i ) (N(v r ) {u r }). Replace uv i with {uv r, v i v }. Case 3 v 1,..., v r 1 N(v r ) ad v i v i for some i ad j with i < j < r. Case 1 applies uless v i, v j N(v r ). Sice d(v i ) d(v i ) > d(v r ), there exists u N(v i ) (N(v r ) {v r }) ad w N(v j ) (N(v r ) {v r }) (possibly u = w). Sice u, w N(v r ), Case 1 applies uless u, w S. Replace {v i v j, uv r } with {uv r, v, v r }. If oe of these case apply, the v 1,..., v r are pairwise adjacet, ad d(v ) = mi{r, d } for > r. Sice S is idepedet, r d(v i) = r(r 1)+ =r+1 mi{r, d }. By (2.4.1), r d 1 is bouded by the right side. Hece we have already elimiated the deficiecy at vertex r. Icrease r by 1 ad cotiue. Tripathi ad Vijay [245] have show that the Erdos-Gallai coditio characterisig graphical degree sequeces of legth eeds to be checed oly for as may as there are distict terms i the sequece ad ot for all, 1. 2.3 Degree Set of a Graph The set of distict o-egative itegers occurrig i a degree sequece of a graph is called its degree set. For example, let the degree sequece be D = [2, 2, 3, 3, 4, 4], the degree set is {2, 3, 4}. A set of distict o-egative itegers is called a degree set if it is the degree set of some graph ad the graph is said to realise the degree set. Let S = {d 1, d 2,..., d } be the set of distict o-egative itegers. Clearly, S is the degree set as the graph G = K d1 +1 K d2 +1... K d +1,
50 Degree Sequeces realises S. This graph has d 1 + d 2 +...+ d + vertices. Example Let S = {1, 3, 4}. The G = K 2 K 4 K 5 (Fig. 2.9). Fig. 2.9 The followig result is due to Kapoor, Polimei ad Wall [126]. Theorem 2.5 Ay set S of distict positive itegers is the degree set of a coected graph ad the miimum order of such a graph is M + 1, where M is the maximum iteger i the set S. Proof Let S be a degree set ad 0 (S) deote the miimum order of a graph G realisig S. As M is the maximum iteger i S, therefore i G there is a vertex adjacet to M other vertices, i.e., 0 (S) M + 1. Now, if there exists a graph of order M + 1 with S as degree set, the 0 (S) = M + 1. The existece of such a graph is established by iductio o the umber of elemets p of S. Let S = {a 1, a 2,..., a p } with a 1 < a 2 <... < a p. For p = 1, the complete graph K a1 +1 realises {a 1 } as degree set. For p = 2, we have S = {a 1, a 2 }. Let G = K a1 VK a2 a 1 +1 (joi of two graphs). Here every vertex of K a1 has degree a 2 ad every other vertex has degree a 1 ad therefore G realises {a 1, a 2 } (Fig. 2.10(a)). For p = 3, we have S = {a 1, a 2, a 3 }. The G = K a1 V(K a3 a 2 H), where H is the graph realisig the degree set {a 2 a 1 } with a 2 a 1 +1 vertices, realises {a 1, a 2, a 3 } (Fig. 2.10 (b)). (Note that d(u) = a 1 1+a 3 a 2 + a 2 a 1 + 1 = a 3, d(v) = a 1, d(w) = a 2 a 2 + a 1 = a 2 ).
Graph Theory 51 Fig. 2.10 Let every set with h positive itegers, 1 h, be the degree set. Let S 1 = {b 1, b 2,..., b +1 } be a ( + 1) set of positive itegers arraged i icreasig order. By iductio hypothesis, there is a graph H realisig the degree set {b 2 b 1, b 3 b 1,..., b b 1 } with order b b 1 +1. The graph G = K b1 V(K b+1 b H), with order b +1 + 1 realises S 1 (Fig. 2.10 (c)). Clearly by costructio, all these graphs are coected. Hece the result follows by iductio. Note that d(u i ) = b 1 1+b +1 b +b b 1 +1 = b +1, d(v i ) = b 1, d(w i )= b i+1 b 1 +b 1 = b i+1, that is d(w 1 ) = b 2, d(w 2 ) = b 3,..., d ( w b b 1 +1) = b b 1 + b 1 = b. Some results o degree sets i bipartite ad tripartite graphs ca be see i [262]. 2.4 New Criterio We have the followig otatios. Let D = [d i ] 1 be a o-decreasig sequece of oegative itegers with 0 d i 1 for all i. Let p 1 be the greatest iteger, p 1 p 2, the secod greatest iteger ad r=1 p r, the th greatest iteger i D, 1 p r (r 1). Let the umber of times the th greatest iteger appears i D be deoted by a. Also, we tae ( ) t = p r = p r, 1 p r (r 1) ad j = 1, 2,..., p +1. r=1 r=1 The followig result due to Pirzada ad YiJia [208] is aother criterio for a oegative sequece of itegers i o-decreasig order to be the degree sequece of some graph. Theorem 2.6 A o-decreasig sequece [d i ] 1 of o-egative itegers, where d i is eve ad 0 d i 1 for all i, is a degree sequece of a graph if ad oly if» t + j 1 d i { j +( m)} a m (2.6.1)
52 Degree Sequeces for all t + j 1+ a m. Note I the above criterio, the iequalities (2.6.1) are to be checed oly for t + j 1+ a m (but ot for greater tha ). We ow illustrate the theorem with the help of the followig examples. Example 1 Let D = [1, 2, 2, 4, 6, 6, 6, 7, 8, 8]. Here, = 10, a 1 = 2, a 2 = 1, a 3 = 3, a 4 = 1, p 1 = 2, p 2 = 1, p 3 = 1, p 4 = 2, so t 1 = 2, t 2 = 3, t 3 = 4, t 4 = 6. ad ad Also, j 1 = 1, j 2 = 1, j 3 = 1, 2. Now, for j 1 = 1, t 1+ j 1 1 d i = 2+1 1 d i = 2 d i = 1+2 = 3, [ j +( m)] a m = 1 [ j 1 +(1 m)]a m = j 1 a 1 = 2. So iequalities (2.6.1) hold. For j 2 = 1, t 2+ j 2 1 d i = 3+1 1 d i = 3 d i = 5 [ j +( m)] a m = 2 [ j 2 +(2 m)]a m = 2a 1 + a 2 = 4+1 = 5. So iequalities (2.6.1) hold. For j 3 = 1, t 3+ j 3 1 d i = 4+1 1 d i = 4 d i = 9 ad 3 [ j 3 +(3 m)]a m = 3 [1+(3 m)]a m = 3a 1 + 2a 2 + a 3 = 6+2+3 = 11. Sice the iequalities (2.6.1) do ot hold (as 9 > 11 is ot true), D is ot the degree sequece. Example 2 Let D = [1, 2, 3, 4, 5, 6, 6, 7, 8, 8]. Here, = 10, a 1 = 2, a 2 = 1, a 3 = 2, a 4 = 1, p 1 = 2, p 2 = 1, p 3 = 1, p 4 = 1, p 5 = 1. So t 1 = 2, t 2 = 3, t 3 = 4, t 4 = 5. Also, j 1 = 1, j 2 = 1, j 3 = 1, j 4 = 1.
Graph Theory 53 For j 1 = 1, t 1+ j 1 1 d i = 2+1 1 d i = 2 d i = 3, ad 1 [ j 1 +(1 m)]a m = a 1 = 2. Obviously the iequalities (2.6.1) hold. For j 2 = 1, t 2+ j 2 1 d i = 3+1 1 d i = 3 d i = 6 ad 2 [ j 2 +(2 m)]a m = 2 [1+(2 m)]a m = 2a 1 + a 2 = 4+1 = 5. Here agai the iequalities (2.6.1) hold. For j 3 = 1, t 3+ j 3 1 d i = 4+1 1 d i = 4 d i = 10 ad 3 [ j 3 +(3 m)]a m = 3 [1+(3 m)]a m = 3a 1 + 2a 2 + a 3 = 6+2+2 = 10. Therefore the iequalities (2.6.1) hold. For j 4 = 1, t 4 + j 4 1 = 5 + 1 1 = 5 ad a 1 + a 2 + a 3 + a 4 = 2 + 1 + 2 + 1 = 6, therefore t 4 + j 4 1+ 4 doe. a m = 5+6 = 11 > 10 ad o further verificatio of the iequalities is to be Hece D is the degree sequece. 2.5 Equivalece of Seve Criteria We list the seve criteria for iteger sequeces to be graphic. A. The Ryser Criterio (Body ad Murty [36] ad Ryser [227]) A sequece [a 1,..., a p ; b 1,..., b ] is called bipartite-graphic if ad oly if there is a simple bipartite graph such that oe compoet has degree sequece [a 1,..., a p ] ad the other oe has [b 1,..., b ]. Defie f = max{i : d i i} ad d 1 = d i + 1 if i f (= {1,..., f }) ad d 1 = d i otherwise. The criterio ca be stated as follows. The iteger sequece [ d 1,..., d ; d 1,..., d ]is bipartite-graphic. (A) B. The Berge Criterio (Berge [23]) Defie [ d 1,..., d ] as follows: For i, d i is the ith colum sum of the (0, 1) matrix, which has for each ad d leadig terms i row
54 Degree Sequeces equal to 1 except for the (, )th term that is 0 ad also the remaiig etries are 0. If d 1 = 3, d 2 = 2, d 3 = 2, d 4 = 2, d 5 = 1, the d 1 = 4, d 2 = 3, d 3 = 2, d 4 = 1, d 5 = 0, ad the (0, 1) matrix becomes 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 The criterio is d i d i for each. (B) C. The Erdos-Gallai Criterio. (Body ad Murty [36]) d i ()( 1)+ mi{, d j } for each. j=+1 (C) D. The Fulerso-Hoffma-McAdrew Criterio (Fulerso[83] ad Grubaum [92) d i ()( m 1)+ i= m+1 d i for each, m 0 ad +m. (D) E. The Bollobas Criterio (Bollabas[29])) d i d i + mi{d j, 1} for each. j=+1 (E) F. The Grubaum Criterio (Grubaum [92]). max{ 1, d i } ()( 1)+ i=+1 d i for each. (F) G. The Hasselbarth Criterio (Hasselbarth [111]) Defie [d i,..., d ] as follows. For i, d i is the ith colum sum of the (0, 1)-matrix i which the d i leadig terms i row i are 1 s ad the remaiig etries are 0 s. The criterio is d i (di 1) for each f, (G) with f = max{i : d i i}.
Graph Theory 55 The followig result due to Siersma ad Hoogevee [235] gives the equivalece amog the above seve criteria. Theorem 2.7 (Siersma ad Hoogevee [235]) Let [d 1,..., d ] be a positive iteger sequece with eve sum. The each of the criteria (A) (G) is equivalet to the statemet that [d 1,..., d ] is graphic. Proof Refer to Ryser [227]. 2.6 Siged Graphs A siged graph is a graph i which every edge is labelled with a + or a. A edge uv labelled with a + is called a positive edge, ad is deoted by uv +. A edge uv labelled with a is called a egative edge, ad is deoted by uv. I a siged graph G(V, E), the positive degree of a vertex u is deg + (u) = {uv : uv + E}, the egative degree of a vertex u is deg (u) = {uv : uv E}, the siged degree of u is sdeg(u) = deg + (u) deg (u) ad the degree of u is deg(u) = deg + (u)+deg (u). A edge uv labelled with a + is called a positive edge, ad is deoted by uv +. A edge uv labelled with a is called a egative edge, ad is deoted by uv. A itegral sequece [d i ] 1 is the siged degree sequece of a siged graph G = (V, E) with V = {v 1, v 2,..., v } if s deg(v i ) = d i, for 1 i. Chartrad et al. [50] have give the characterisatio of siged degree sequeces of siged paths, siged stars, siged double stars ad complete siged graphs. A itegral sequece is s-graphical if it is the siged degree sequece of a siged graph. A itegral sequece [d i ] 1 is stadard if 1 d 1 d 2... d ad d 1 d. The followig lemma shows that a siged degree sequece ca be modified ad rearraged ito a equivalet stadard form. Lemma 2.1 If [d i ] 1 is the siged degree sequece of a siged graph G, the [ d i] 1 is the siged degree sequece of the siged graph G obtaied from G by iterchagig positive edges with egative edges. The followig ecessary ad sufficiet coditio uder which a itegral sequece is s-graphical is due to Chartrad et al. [50]. Theorem 2.8 A stadard itegral sequece [d i ] 1 is s-graphical if ad oly if the sequece [d 2 1, d d1 +s+1 1, d d1 +s+2,..., d s, d s+1 + 1,..., d + 1] is s-graphical for some 0 s ( 1 d 1 )/2. Remar We ote that Haimi s theorem for degree sequeces is a case of Theorem 2.8 by taig s = 0. This leads to a efficiet algorithm for recogisig the degree sequeces of a graph. But the wide degree of latitude for choosig s i Theorem 2.8 maes it harder to devise a efficiet algorithm implemetatio.
56 Degree Sequeces The followig result due to Ya et al. [271] provides a good choice for parameter s i Theorem 2.8. It leads to a polyomial time algorithm for recogisig siged degree sequeces. Theorem 2.9 A stadard sequece D = [d i ] 1 is s-graphical if ad oly if D m = [d 2 1, d d1 +m+1 1,..., d d1 +m+2,..., d m, d m+1 + 1..., d + 1] is s-graphical, where m is the maximum o-egative iteger such that d d1 +m+1 > d m+1. Proof Let D be the siged degree sequece of a siged graph G = (V, E) with V = {v 1, v 2,..., v } ad sdeg(v i ) = d i, for 1 i. For each s, 0 s ( 1 d 1 )/2, cosider the sequece D s = [d 2 1,..., d d1 +s+1 1, d d1 +s+2,..., d s, d s+1 + 1,..., d + 1]. By Theorem 2.8, D s is s-graphical for some s. We may choose s such that s m is miimum. Suppose G = (V, E ) is a siged graph with V = {v 2, v 3,..., v } whose siged degree sequece is D s. If s < m, the d a > d b by the choice of m, where a = d 1 +s+2 ad b = s. Sice d a > d b, there exists some vertex v of G differet from v a ad v b ad satisfies oe of the followig coditios. i. v a v + is a positive edge ad v bv is a egative edge. ii. v a v + is a positive edge ad v b is ot adjacet to v iii. v a is ot adjacet to v ad v b v is a egative edge For (i), remove v a v + ad v b v to G, ad for (ii), remove v a v + from G ad add a ew positive edge v b v + to G ad for (iii), remove v b v from G ad a ew egative edge v a v to G. These modificatios result i a siged graph G whose siged degree sequece D s+1. This cotradicts the miimality of s m. If s > m, the d d1 +s+1 = d s+1, ad therefore, d d1 +s+1 1 < d s+1 1. A argumet similar to the above leads to a cotradictio i the choice of s. Therefore, s = m ad D m is s-graphical. Coversely, suppose D m is the siged degree sequece of a siged graph G = (V, E ) i which V = {v 2, v 3,..., v }. If G is the siged graph obtaied from G by addig a ew vertex v 1 ad ew positive edges v 1 v + i for 2 i d 1 + m + 1 ad ew egative edges v 1 v j for m+1 j, the D is the siged degree sequece of G. I a siged graph G = (V, E) with V =, E = m, we deote by m + ad m respectively, the umbers of positive edges ad egative edges of G. Further, +, 0 ad deote respectively, the umbers of vertices with positive, zero ad egative siged degrees. The followig result is due to Chartrad et al. [50]. Lemma 2.2 If G = (V, E) is a siged graph with V =, E = m, the = s deg(v) v V 2m(mod4), m + = 1 4 (2m+) ad m = 1 4 (2m ).
Graph Theory 57 The ext result is due to Ya et al [271]. Lemma 2.3 2 0 2m. For ay siged graph G = (V, E) without isolated vertices, sdeg(v) + v V Proof First, each sdeg(v) = deg + (v) deg (v) deg + (v)+ deg (v). Sice G has o isolated vertices, 2 deg + (v)+ deg (v) whe sdeg(v) = 0. Thus, s deg (v) + 2 0 (deg + (v)+deg (v) = 2m + + 2m = 2m. v V v V Lemma 2.4 For ay coected siged graph G =(V, E), s deg (v) +2 sdeg(v) v V sdeg(v)<0 6m+4 4α 4 + 4 0, where α = 1 if + > 0 ad α = 0 otherwise. Proof Cosider the subgraph G = (V, E ) of G iduced by those edges icidet to vertices with o-egative siged degrees. We have, sdeg(v) 2 (umber of positive edges i G ) sdeg(v)>0 (umber of egative edges i G ) 3m + E. Sice G is coected, each compoet of G cotais at least oe vertex of egative siged degree except for the case of G = G. Therefore, + + 0 1+α E. Thus, ( ) sdeg(v) + + + 0 1+α 3m + 1 = 3 2 m+ 1 4 sdeg(v). sdeg(v).0 v V Hece, sdeg(v) + 2 v V sdeg(v)<0 sdeg(v) 6m+4 4α 4 + 4 0. For ay iteger, copies of v i v j meas copies of positive edges v i v + j if > 0, o edges if = 0 ad copies of egative edges v i v j if < 0. The ext result for siged graphs with loops or multiple edges is due to Ya et al. [271]. Theorem 2.10 ad oly if d i is eve. A itegral sequece [d i ] 1 is the siged degree sequece of a siged if Proof The ecessity follows from Lemma 2.2. Sufficiecy Let d i be eve. The the umber of odd terms is eve, say d i = 2e i + 1 for 1 i 2 ad d i = 2e i for 2+1 i p. The [d 1, d 2,..., d ] is the siged degree sequece
58 Degree Sequeces of the siged graph with vertex set {v 1, v 2,..., v } ad edge set { d 3 = 1 2 v 1 v 2 } {d 2 + d 3 1 2 v 3 v i : 4 i }. d i copies of v 2 v 3 } {d 1 + d 3 1 2 d i copies of d i copies of v 1 v 3 } {d i copies of Various results o siged degrees i siged graphs ca be foud i [259], [263], [264] ad [266]. 2.7 Exercises 1. Verify whether or ot the followig sequeces are degree sequeces. a. [ 1, 1, 1, 2, 3, 4, 5, 6, 7], b. [ 1, 1, 1, 2, 2, 2], c. [ 4, 4, 4, 4, 4, 4], d. [ 2, 2, 2, 2, 4, 4]. 2. Show that there is o perfect degree sequece. 3. What coditios o ad will esure that is a degree sequece? 4. Give a example of a graph that ca ot be geerated by the Wag-Kleitma algorithm. 5. Draw the five o isomorphic graphs with degree sequece [3, 3, 2, 2, 1, 1]. 6. Show that a graph ad its complemet have the same frequecy sequece. 7. Costruct a graph with a degree sequece [3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1] by usig Havel-Haimi algorithm.