Graphs and Combinatorics最新文献

In the year 2000, Eric Egge introduced the generalized Terwilliger algebra ({mathcal {T}}) of a distance-regular graph (varGamma ). For any vertex x of (varGamma ), there is a surjective algebra homomorphism (natural ) from ({mathcal {T}}) to the Terwilliger algebra T(x). If (varGamma ) is a complete graph, then (natural ) is an isomorphism. If (varGamma ) is not complete, then (natural ) may or may not be an isomorphism, and in general the details are unknown. We show that if (varGamma ) is a hypercube, there exists an isomorphism from ({mathcal {T}}) to a direct sum of full matrix algebras. Using this result, we then show that if (varGamma ) is a hypercube, the algebra homomorphism (natural :{mathcal {T}}rightarrow T(x)) is an isomorphism for all vertices x of (varGamma ).

The DP-coloring is a generalization of the list coloring, introduced by Dvořák and Postle. Let ({mathcal {H}}=(L,H)) be a cover of a graph G and (P_{DP}(G,{mathcal {H}})) be the number of ({mathcal {H}})-colorings of G. The DP color function (P_{DP}(G,m)) of G, introduced by Kaul and Mudrock, is the minimum value of (P_{DP}(G,{mathcal {H}})) where the minimum is taken over all possible m-fold covers ({mathcal {H}}) of G. For the family of n-vertex connected graphs, one can deduce that trees maximize the DP color function, from two results of Kaul and Mudrock. In this paper we obtain tight upper bounds for the DP color function of n-vertex 2-connected graphs. Another concern in this paper is the canonical labeling in a cover. It is well known that if an m-fold cover ({mathcal {H}}) of a graph G has a canonical labeling, then (P_{DP}(G,{mathcal {H}})=P(G,m)) in which P(G, m) is the chromatic polynomial of G. However the converse statement of this conclusion is not always true. We give examples that for some m and G, there exists an m-fold cover ({mathcal {H}}) of G such that (P_{DP}(G,{mathcal {H}})=P(G,m)), but ({mathcal {H}}) has no canonical labelings. We also prove that when G is a unicyclic graph or a theta graph, for each (mge 3), if (P_{DP}(G,{mathcal {H}})=P (G,m)), then ({mathcal {H}}) has a canonical labeling.

We first give a proof of the Alon–Tarsi list coloring theorem that differs from Alon and Tarsi’s original. We use the ideas from this proof to obtain the following result, which is an additive coloring analog of the Alon–Tarsi Theorem: Let G be a graph and let D be an orientation of G. We introduce a new digraph (mathcal {W}(D)), such that if the out-degree in D of each vertex v is (d_v), and if the number of Eulerian subdigraphs of (mathcal {W}(D)) with an even number of edges differs from the number of Eulerian subdigraphs of (mathcal {W}(D)) with an odd number of edges, then for any assignment of lists L(v) of (d_v+1) positive integers to the vertices of G, there is an additive coloring of G assigning to each vertex v an element from L(v). As an application, we prove an additive list coloring result for tripartite graphs G such that one of the color classes of G contains only vertices whose neighborhoods are complete.

Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.

Let G be a graph. A bisection of G is a bipartition of V(G) with (V(G)=V_1cup V_2), (V_1cap V_2=emptyset ) and (||V_1|-|V_2||le 1). Bollobás and Scott conjectured that every graph admits a bisection such that for every vertex, its external degree is greater than or equal to its internal degree minus one. In this paper, we confirm this conjecture for some regular graphs. Our results extend a result given by Ban and Linial (J Graph Theory 83:5–18, 2016). We also give an upper bound of the maximum bisection of graphs.

Let V be an n-dimensional vector space over the finite field ({mathbb {F}}_{q}) and let (left[ begin{array}{c} V k end{array}right] _q) denote the family of all k-dimensional subspaces of V. A family ({{mathcal {F}}}subseteq left[ begin{array}{c} V k end{array}right] _q) is called intersecting if for all F, (F'in {{mathcal {F}}},) we have ({textrm{dim}}(Fcap F')ge 1.) A family ({{mathcal {F}}}subseteq left[ begin{array}{c} V k end{array}right] _q) is called almost intersecting if for every (Fin {{mathcal {F}}}) there is at most one element (F'in {{mathcal {F}}}) satisfying ({textrm{dim}}(Fcap F')=0.) In this paper we investigate almost intersecting families in the vector space V. Firstly, for large n, we determine the maximum size of an almost intersecting family in (left[ begin{array}{c} V k end{array}right] _q,) which is the same as that of an intersecting family. Secondly, we characterize the structures of all maximum almost intersecting families under the condition that they are not intersecting.

We use the reciprocal transformation to propose the closed-form solutions to the conics through m points and tangent to n lines satisfying (m+n=5) in general position. We also derive the algebraic and geometric necessary and sufficient conditions for the non-degenerate real conics.