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Register a new userGenerating families of surface triangulations. The case of punctured surfaces with inner degree at least 4
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Mar\\'ia Trinidad Villar-Li\\~n\\'an
Publication date
2015
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English
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A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on $S$, and we list the possibilities. A key ingredient of the proof is to show that any two $d$-regular triangulations of the plane for $d> 6 $ are combinatorially equivalent. The proof of this uniqueness result, which is of independent interest, is based on an inductive argument involving some combinatorial topology.
Let $G$ be an $n$-vertex graph and let $L:V(G)rightarrow P({1,2,3})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can determine $L$-choosability of $G$ in $O(1.3196^{n_3+.5n_2})$ time, where $n_i$ is the number of vertices in $G$ with list of size $i$ for $iin {2,3}$. As a corollary, we conclude that the 3-colorability of any graph $G$ with minimum degree at least 6 can be determined in $O(1.3196^{n-.5Delta(G)})$ time.
Let $G$ be an $n$-vertex graph with the maximum degree $Delta$ and the minimum degree $delta$. We give algorithms with complexity $O(1.3158^{n-0.7~Delta(G)})$ and $O(1.32^{n-0.73~Delta(G)})$ that determines if $G$ is 3-colorable, when $delta(G)geq 8$ and $delta(G)geq 7$, respectively.
The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $mathscr D$ is $1+max mathscr D$. Tripathi & Vijay considered the analogous problem concerning the least size of graphs with degree set $mathscr D$. We expand on their results, and determine the least size of graphs with degree set $mathscr D$ when (i) $min mathscr D mid d$ for each $d in mathscr D$; (ii) $min mathscr D=2$; (iii) $mathscr D={m,m+1,ldots,n}$. In addition, given any $mathscr D$, we produce a graph $G$ whose size is within $min mathscr D$ of the optimal size, giving a $big(1+frac{2}{d_1+1})$-approximation, where $d_1=max mathscr D$.
A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $iin {1, ..., r}$. For a given pair $(g, d_1)$, the question of determining the minimum $d_2=d_2(g; d_1)$ such that planar graphs with girth at least $g$ are $(d_1, d_2)$-colorable has attracted much interest. The finiteness of $d_2(g; d_1)$ was known for all cases except when $(g, d_1)=(5, 1)$. Montassier and Ochem explicitly asked if $d_2(5; 1)$ is finite. We answer this question in the affirmative with $d_2(5; 1)leq 10$; namely, we prove that all planar graphs with girth at least $5$ are $(1, 10)$-colorable. Moreover, our proof extends to the statement that for any surface $S$ of Euler genus $gamma$, there exists a $K=K(gamma)$ where graphs with girth at least $5$ that are embeddable on $S$ are $(1, K)$-colorable. On the other hand, there is no finite $k$ where planar graphs (and thus embeddable on any surface) with girth at least $5$ are $(0, k)$-colorable.
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