O (7), T(3), F(10) 

conjectures 39 and, 47 were repeated with conjectures 57-67.

note: conjecture 5 was repeated from the first run for L.

maximum { dist_even(v) :  dist_even(v) is the number of vertices at even distance from v }  ≤ 1 + σ(G),

then G has a Hamiltonian path.

If G is a simple connected graph with n > 1 such that [n - g_c+ 1]/2 ≤ σ(G), then G has a Hamiltonian path.

It is not difficult to see that FLOOR(n(G)/2) £ annihilation(G) and it is known that if minimum degree is at least 3 then g_t(G) ≤ n(G)/2. Thus, if minimum degree is at least three then g_t(G) ≤ annihilation(G). From this we see that the only unsettled case is when minimum degree is at most 2, which includes trees. Note that for G a tree c_C4(G) = 1, and the conjecture restricted to trees becomes g_t(G) ≤ annihilation(G)+ 1 if G is a tree.

Nov 2012, Desormeaux, Haynes and Henning proved that if G is a tree on n(G)³ 2, then g_t(G) ≤ annihilation(G)+ 1, see [DHH].

[DHH] Wyatt J. Desormeaux, Teresa W. Haynes and Michael A. Henning, Relating the annihilation number and the total domination number of a tree, Discrete Applied Mathematics, article in press available online Oct 2012.

where S(T) is the set of support vertices, L is the set of leaves, and D₂={v| deg(v) = 2}.

       γ₂ ≤ -1 + a*(d + |A₁|) ,

where d is the minimum degrees and A₁ is the set of vertices that are adjacent to exactly one non-minimum degree vertex.

       γ₂ ≤ [a*ecc_avg(A)+ |B|]/2,

where A is the set of minimum degree vertices and B is the set of periphery vertices.

A special case of this would suggest a sufficient condition for γ₂ ≤ a + |B|/2. Namely, if the minimum degree vertices have eccentricity two, then γ₂ ≤ a + |B|/2. If  the minimum degree vertices have eccentricity at least four then this follows from γ₂ ≤ 2a.

       γ₂ ≤ [diam(G)(a+ |X|)]/2,

where X is the set of vertices whose maximum local independence is maximum.

A special case of this would suggests that if the diameter of G is two, then γ₂ ≤ a + |X|/2. If  the diameter at least four then this follows from   γ₂ ≤ 2a.

       γ₂ ≤ Maxine + γ.       

          γ₂ ≤ n - m + 1.   

where m is the (upper) median of the degree sequence.

          γ₂ ≤ ≤ p(G) + 1 + m(G[A₂]) + |E(G[T])|,

where A is the set vertices of degree two and T is the set of vertex of degree at least three.

       γ₂ ≤ -1 + a*(d + |A₁|) ,

where d is the minimum degrees and A₁ is the set of vertices that are adjacent to exactly one non-minimum degree vertex.

       γ₂ ≤ [a*ecc_avg(A)+ |B|]/2,

where A is the set of minimum degree vertices and B is the set of periphery vertices.

A special case of this would suggest a sufficient condition for γ₂ ≤ a + |B|/2. Namely, if the minimum degree vertices have eccentricity two, then γ₂ ≤ a + |B|/2. Here is an example of a graph for which the conjectured relation is sharp, there is one minimum degree vertex of eccentricity 2, d=3, |B| = 2, a=3, and γ₂ = 4. If the average of the eccentricities of the minimum degree vertices is at least four then this follows from γ₂ ≤ 2a.

       γ₂ ≤ [diam(G)(a+ |X|)]/2,

where X is the set of vertices whose local independence is maximum.

A special case of this would suggests that if the diameter of G is two, then γ₂ ≤ a + |X|. Here is an example of a graph for which the conjectured relation is sharp,  diam(G) = 2, |X| = 1, a=3, and γ₂ = 4. If  the diameter at least four then this follows from γ₂ ≤ 2a.

       γ₂ ≤ Maxine + γ.       

          γ₂ ≤ FLOOR(n - |T |/2),

where T = {k : some vertex of G has exactly k triangles incident to it}.   

          γ₂ ≤ n - m + 1.   

where m is the (upper) median of the degree sequence.

          γ₂ ≤ |M|*(q + |A₂|) -1,

where A₂  is the set vertices of degree at most two, M is the set of vertices of mode_min degree and q is the 1st quartile degree .

          γ₂ ≤  p(G) + 1 + m(G[T]) + |E(G[A₃])|,

where T is the set vertices of degree two and A₃ is the set of vertices of degree at least three.

          γ₂ ≤  m(G[V - A_d ]) + |A_d | + |{v : |N(v) Ç A₂ | = 1}| ,

where A_d  is the set of vertices of minimum degree and A₂ is the set of vertices of degree 2.

          γ₂ ≤  m(G[V - A_d ]) + |A_d | + k_v(G),

where A_d  is the set of vertices of minimum degree and k_v is the number of cut vertices.

          γ₂ ≤  m(G) + m(G[V - N(N(P))]) + |A_d |,

where A_d  is the set of vertices of minimum degree and P is the set of pendants.

          γ₂ ≤  m(G) + m(G[N(A₂)]) + |A_d |,

where A_d  is the set of vertices of minimum degree and  A₂ is the set vertices of degree two.

          γ₂ ≤  m(G) + m(G[N(A₂)]) + |{v : |N(v) Ç N(A) | = 1}|,

where A is the set of vertices of minimum degree and  A₂ is the set vertices of degree two.

          γ₂ ≤  | V - N(N(P))| + m(G[N(N(P))])  + |P |,

where P is the set of pendants.

          γ₂ ≤  | V - N(N(P))| + |E(G[N(P)]) | + |{v : |N(v) Ç N(P) | = 1}|,

where P is the set of pendants.

          γ₂ ≤  | V - N(N(P))| + |N(P) | + |{v : |N(v) Ç N[P] | = 1}|,

where P is the set of pendants.

          γ₂ ≤   µ(G[V - N(N(P))]) + a (G[V-M]) + |M₄|,

where P is the set of pendants, M is the set of maximum degree vertices and M₄ is the set of vertices each of which is incident with the maximum number of K₄.

          γ₂ ≤  m(G) + |E(G[A₂])| + |M |,

where M is the set of vertices of min local independence and  A₂ is the set vertices of degree two.

          γ₂ ≤  |M| + m(G[V-A_d ]) + m(G[N(A₂)]),

where M is the set of vertices of mode_min degree, A_d  is the set of minimum degree vertices and  A₂ is the set vertices of degree two.

          γ₂ ≤  |M |+ d(G[V-A]) + |V-A₃|,

where M is the set of vertices of mode_min degree, A is the set of minimum degree vertices and  A₃ is the set vertices of degree at least three.

          γ₂ ≤  dd_mode + |M | + |E(A₃, V-A₃)|,

where M is the set of vertices of mode_min degree and  A₃ is the set vertices of degree at least three.

         γ₂ ≤  |V - N(P)| + γ(G²),

where P is the set of pendants.

         γ₂ ≤  |E(G[V - NN((P))])| + |{v : |N(v) Ç [V - N(P)] | = 1}| + |E(P,V-P)| ,

where P is the set of pendants.

         γ₂ ≤ WP(G) + FLOOR[a (G[C])/2],

where C is the center.

*May 2012, in [DP12] we proved that a₂ ≤ WP(G) + 1, so here (since  γ₂ ≤ a₂ ) it simply remains to settle γ₂ ≤ WP(G) whenever the a (G[C]) = 1.

[DP12] E. DeLaVina, R. Pepper, Graffiti.pc on the 2-independence number of a graph, preprint 2012.

         γ₂ ≤ WP(G) + FLOOR[3/a (G[A₃])],

where  A₃ is the set vertices of degree at least three.

*May 2012, in [DP12] we proved that a₂ ≤ WP(G) + 1, so here (since  γ₂ ≤ a₂ ) it simply remains to settle γ₂ ≤ WP(G) whenever the a (G[A₃]) ³ 4.

[DP12] E. DeLaVina, R. Pepper, Graffiti.pc on the 2-independence number of a graph, preprint 2012.

         γ₂ ≤ (2/3)WP(G) + 2|M|,

where  M is the set vertices of minimum local independence.

where  A(G) is the annihilation number .

Feb. 2019, in [DHR14] Desormeaux et. al, proved that for trees the following holds  γ₂ ≤ A(G) + 1.

 γ₂ ≤  1+ FLOOR[Tdist_max/disp_avg].

      γ₂ ≤  FLOOR[3*Tdist_max / freq[T_max(v)]].

      γ₂ ≤  2[isolates(G[A_d ]) + |{v : |N(v) Ç A_D | = 1}| + γ_t ] ,

where A_d  is the set of vertices of minimum degree and A_D  is the set of maximum degree vertices.

 if G is k-regular with k ≤  n/2, then |H| ≤ |E(G[D_e])|?

 if G is k-regular with k ≤  n/2, then |H| ≤ |E(G[D_e])|?

*May 2012, in [DP12] we proved that a_k ≤ WP(G) + k-1, so here it simply remains to settle a₂ ≤ WP(G) whenever the neigbor dominators are isolated in the subgraph that they induce and when k=1.

[DP12] E. DeLaVina, R. Pepper, Graffiti.pc on the 2-independence number of a graph, preprint 2012.